Serre’s conjectures

Kenneth A. Ribet William A. Stein

Contents

Serre’s conjectures 1 Lecture 1. Introduction to Serre’s conjecture 3 Lecture 2. The strong conjecture of Serre 7 Lecture 3. The weight in Serre’s conjecture 11 Lecture 4. Edixhoven’s proof 17 Lecture 5. Galois representations from modular forms 21 Lecture 6. Introduction to level lowering 25 Lecture 7. Mazur’s principle 29 Lecture 8. Level lowering with multiplicity one 33 Lecture 9. Katz modular forms (Appendix) 37 Lecture 10. Exercises (Appendix) 41 Bibliography 49

IAS/Park City Mathematics Series Volume 00, 1999

Serre’s conjectures

Kenneth A. Ribet

These are at present rough notes that I (William Stein) am writing based on Ken Ribet’s notes and lectures. (Ken Ribet wrote this introduction). I haven’t labeled the “exercises” and “appendix” sections correctly. Perhaps they should be put at the end of the appropriate lectures.

My plan is to begin by discussing some examples of mod ℓ representations of Gal(\overline{Q}/Q). I’ll try to motivate Serre’s conjectures by referring first to the case of representations that are unramified outside ℓ; these should come from cusp forms on the full modular group SL_{2}(Z)Z. In another direction, one might think about representations coming from ℓ-division points on elliptic curves, or more generally from ℓ-division points on abelian varieties of “GL_{2}-type.” Amazingly (to me), Serre’s conjectures imply that all odd irreducible two-dimensional mod ℓ representations of Gal(\overline{Q}/Q) may be realized in spaces of ℓ-division points on such abelian varieties. The weak Serre conjecture states that all such representations come from modular forms, and then it takes only a bit of technique to show that one can take the modular forms to have weight two (if one allows powers of ℓ in the level).

Since little work has been done toward proving the weak Serre conjecture, my lectures will focus on the bridge between the weak and the strong conjectures. The latter states that each ρ as above comes from the space of cusp forms of a specific weight and level, with these invariants between determined by the local behavior of ρ at ℓ and at primes other than ℓ (respectively). To motivate the strong conjecture, and to work toward the bridge, we need to discuss the local behavior of those ρ that do come from modular forms. For the most part, we can look only at forms of weight k ≥ 2 whose levels N are prime to ℓ. In that case, the behavior of ρ at ℓ is described in detail in [17], where theorems of Deligne and Fontaine are recalled. (In [17, §6], Edixhoven supplies a proof of Fontaine’s theorem.) Further, the behavior of ρ at primes p ≠ ℓ may be inferred from Carayol’s theorems [5, 6], which relate the behavior at p of the ℓ-adic representations attached to f with the

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

1Math Department, MC 3840; Berkeley, CA 94720-3840.

E-mail address: ribetmath.berkeley.edu.

p-adic component of the automorphic representation of GL(2) that one associates with f. (The behavior of ρ at ℓ in the case where ℓ divides N is analyzed in [35].)

In [40], Serre associates to each ρ a level N(ρ) and a weight k(ρ). These invariants are defined so that N(ρ) is prime to ℓ and so that k is an integer greater than 1. As Serre was no doubt aware, if ρ arises from a modular form of weight k and level N, and if k is at least 2 and N is prime to ℓ, then one has k(ρ) ≤ k and N(ρ) | N. To find an f for which N = N(ρ) and k = k(ρ) is to “optimize” the level and weight of a form giving ρ. As Edixhoven explains in his article [17] weight optimization follows in a somewhat straightforward manner from the theorems of Deligne and Fontaine alluded to above, plus the theorem of Gross on companion forms [20]. (See [9] for a partial treatment of the companion form problem when ℓ = 2.) Luckily, we can perform weight optimization after having optimized the level, since the weight optimization machinery does not disturb the level at any point.

In [6], Carayol analyzes the level optimization problem. He shows, in particular, that the problem breaks down into a series of sub-problems, all but one of which he treats by appealing to a single lemma, the lemma of [6, §3]. The remaining subproblem is the one which intervenes in showing the implication “Shimura–Taniyama ⟹ Fermat.” I have written about this sub-problem repeatedly [33, 32, 34, 47]. Most likely, I will explain the principle of [33] early on in these lectures. Then, toward the end of the series, I will explain the more refined level-lowering argument that I introduced in my article [34]. (This argument was extended by Diamond in [13].) One reason for going over the argument now is that the argument hasn’t yet been made to work for ℓ = 2. Perhaps someone in the audience will see how to modify the argument so that it extends to this case.

A certain amount of work has been done on the Hilbert modular case, i.e., the case where Q is replaced by a totally real number field F. For this work, the reader may consult articles of Frazer Jarvis and of Kazuhiro Fujiwara. I am especially grateful to Fujiwara for his sending me a preliminary version of his manuscript, “Level optimization in the totally real case.” I suspect, however, that these lectures will treat only the classical case F = Q.

Expanded notes for these lectures will be written in collaboration with William A. Stein, the TA for this course. A good interim reference for much of this material is Bas Edixhoven’s article [15].

Background material and basic terminology

Notation

LECTURE 1

Introduction to Serre’s conjecture

A Galois representation is a continuous homomorphism ρ : G = Gal(\overline{Q}/Q) → GL(2, \overline{F}_{ℓ}), where G is the profinite limit of the groups Gal(K/Q) as K varies over finite Galois extensions, G is endowed with the profinite topology, and GL(2, \overline{F}_{ℓ}) is given the discrete topology. The image of ρ is finite. There exists non-continuous homomorphisms with finite image (Exercise 10.7), but we will not consider them in these lectures. The field fixed by ker(ρ) is a finite Galois extension K of Q, and ρ factors through Gal(K/Q), which in turn embeds into some GL(2, F_{ℓ^{ν}}).

Let A be an elliptic curve. Let n ≥ 1 be an integer and consider the group A[n] ≈ (Z/nZ)^{2} of n-torsion points on A(\overline{Q}). After choosing a basis we obtain a homomorphism ρ_{A,n} : G ⟶ GL(2, Z/nZ). With n varying over prime numbers ℓ we may view this as a family of Galois representations. Looking in tables [10] we find many elliptic curves given by equations. For example, the curve

B : y^{2} + y = x^{3} + x^{2} − 12x + 2.

Since B is isolated in its isogeny class, the representations ρ_{B,ℓ} are all irreducible (Exercise 10.3).

The mod n cyclotomic character is

χ_{n} : G → Gal(Q(ζ_{n})/Q) = Aut μ_{n} ≅ (Z/nZ)^{∗}.

The Weil pairing defines an isomorphism ∧^{2}A[n] ≅ μ_{n}, so det(ρ_{A,n}) = χ_{n} (Exercise 10.5). The complex conjugation c ∈ Gal(\overline{Q}/Q) acts on on μ_{n} by z ↦ z^{−1}, so χ_{n}(c) = −1. Thus det ρ_{A,ℓ}(c) = −1. Call a Galois representation odd if the determinant of the action of complex conjugation is −1. If ℓ > 2, then ρ_{A,ℓ}(c) is conjugate over \overline{F}_{ℓ} to ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) (Exercise 10.6 last part).

In 1987 Serre [40] made very precise conjectures about odd irreducible Galois representations ρ which take values in GL(2, \overline{F}_{ℓ}). The reducible case is also of considerable interest. Serre conjectured that if ρ is odd and irreducible, then ρ arises from a modular form via a construction similar to the above. Thus the conjecture asserts that the very condition of being odd, visibly necessary for ρ to come from a modular form, is also sufficient.

The conjecture of Shimura and Taniyama, now a theorem, asserts that there is a bijection between elliptic curves A/Q and rational newforms f. The conductor of A is a positive integer divisible exactly by the primes of bad reduction. The conjecture produces a modular form f ∈ S_{2}(Γ_{0}(N)) whose Dirichlet series

f = ∑_{n=1}^{∞} c_{n}q^{n} = ∑_{n=1}^{∞} c_{n}e^{2πizn}

is connected to A by the relation

∑_{n=1}^{∞} \frac{c_{n}}{n^{s}} = L(f, s) = L(A, s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

The integers a_{n} count the number of points on A over various finite fields.

Fix an elliptic curve A, a prime ℓ, and let ρ_{ℓ} : G → GL(2, F_{ℓ}) be the associated representation. Make the hypothesis that ρ_{ℓ} is irreducible. This is true for all but finitely many ℓ. Mazur [28], showed that if ℓ > 163 then ρ_{ℓ} is irreducible. The arithmetic of this representation is encoded in its values on the Frobenius elements of G. We now recall some facts about ramification. Let N be the conductor of A. If p ∤ ℓN then ρ_{ℓ} is unramified at p, i.e., the finite extension K = Q(E[ℓ]) of Q obtained by adjoining all coordinates of ℓ-torsion points is unramified at p (Exercise 10.13). Let λ be some prime of the ring O_{K} of integers of K lying over p, and let D_{p} be the subgroup of Gal(K/Q) of elements which preserve λ. The natural map D_{p} → Aut(O_{K}/λ) is surjective. Let Frob_{p} ∈ Gal(K/Q) denote a lift of the p-th power automorphism of the finite field O_{K}/λ; it is ill-defined in two ways:

1. Modulo the kernel I_{p}: but p is unramified, so I_{p} is trivial.

2. Modulo choice of prime λ lying over p: but the primes are all conjugate so det(ρ_{ℓ}(Frob_{p})), tr(ρ_{ℓ}(Frob_{p})) ∈ F_{ℓ} are well-defined.

The weak conjecture of Serre asserts that every irreducible odd ρ comes from a modular form f = ∑ a_{n}q^{n} of some weight k ≥ 2 and level N ≥ 1. Warning: even if ρ takes values in GL_{2}(F_{ℓ}), the a_{n} need not lie in Q. The space of modular forms of weight k and level N is denoted S_{k}(Γ_{1}(N)). Define two subgroups of SL_{2}(Z):

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : c ≡ 0 (mod N)\}
Γ_{1}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) : a ≡ d ≡ 1 (mod N)\}

A modular form is a holomorphic function on the upper half plane 𝔥 = {z ∈ C : ℑ(z) > 0} such that

f (\frac{az + b}{cz + d}) = (cz + d)^{k}f(z),

satisfying additional conditions at the “cusps.” For a precise definition see [25, 30, 37, 44]. One of the primary objects associated to S_{k}(Γ_{1}(N)) is a large family of commuting operators,

T_{n}, n ≥ 1, ⟨d⟩, d ∈ (Z/nZ)^{∗},

called the Hecke operators, which operate on this space. The ring

T = Z[… T_{n} …, … ⟨d⟩ …] ⊂ End(S_{k}(Γ_{1}(N))

is called the Hecke algebra. Suppose 0 ≠ f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{1}(N)) is an eigenform for all the T_{n} and ⟨d⟩. We find that if T_{n}f = c_{n}f then a_{n} = a_{1}c_{n}. If a_{1} = 0, then f = 0, so a_{1} ≠ 0. Normalize f so that a_{1} = 1. Then f “wears its eigenvalues on its sleave” so to speak: f = ∑ c_{n}q^{n}. Henceforth we will assume that all eigenforms are normalized.

What does the conjecture of Serre assert? Starting with a representation ρ, Serre conjectures that there exists k, N, f such that tr(Frob_{p}) and det(Frob_{p}) are encoded by the eigenvalues of the eigenform f ∈ S_{k}(Γ_{1}(N)). One problem is that tr(Frob_{p}) and det(Frob_{p}) lie in \overline{F}_{ℓ}, whereas the eigenvalues a_{n} lie in C. The field

E = E_{f} = Q(… a_{n} …)

is a finite extension of Q equipped with a canonical conjugation, so E is either totally real, or a quadratic imaginary extension of a totally real field. Furthermore, the a_{n} lie in the ring O of integers of E. Let φ : O → \overline{F}_{ℓ} be a homomorphism. To give φ is to give a maximal ideal λ = ker(φ) of O having residue characteristic ℓ, and an embedding of O/λ in \overline{F}_{ℓ}. We can now be more precise. Say that ρ arises from a modular form, or is modular, if there exists k, N, f, and φ : O → \overline{F}_{ℓ} such that, for all p ∤ ℓN,

tr(ρ(Frob_{p})) = φ(a_{p})
det(ρ(Frob_{p})) = p^{k−1}φ(ε(p)),

where ε : (Z/NZ)^{∗} → O^{∗} gives the action of ⟨d⟩ on the eigenform f. Note that since Gal(Q(μ_{N})/Q) ≅ (Z/NZ)^{∗}, we may view ε as a character of Gal(\overline{Q}/Q).

Where does Serre’s conjecture come from? Given an eigenform f ∈ S_{k}(Γ_{1}(N)) one can construct a Galois representation. Deligne constructed representations attached to the eigenform Δ of weight 12 and level 1 in a conscise Bourbaki article [11]. In fact, he constructs λ-adic representations ρ_{λ} where λ is a prime of O lying over ℓ:

\xymatrix{G \ar[r]^{ρ_{λ}} & GL(2, O_{λ}) \ar[d] & GL(2, \overline{F}_{ℓ}) \\  & GL(2, O/λ) \ar@{^{(}->}[ur]^{φ}}

The representation ρ_{λ} is unramified at all p ∤ ℓN and for such p,

tr(ρ_{λ}(Frob_{p})) = a_{p} ∈ O_{λ}
det(ρ_{λ}(Frob_{p})) = ε(p)p^{k−1} ∈ O_{λ}

What about the oddness condition in Serre’s conjecture? Let f ∈ S_{k}(Γ_{1}(N)) be a newform and ρ the associated representation. By Exercise 10.6, ε(−1) = (−1)^{k}, so

det(ρ(c)) = χ^{k−1}(−1) · ε(−1) = (−1)^{k−1}(−1)^{k} = (−1)^{k+k−1} = −1.

LECTURE 2

The strong conjecture of Serre

Let ρ : G = Gal(\overline{Q}/Q) ⟶ GL(2, \overline{F}_{ℓ}) be an odd irreducible Galois representation. Recall that the weak conjecture asserts that ρ is modular, i.e., there exists N, k such that ρ comes from some f ∈ S_{k}(Γ_{1}(N)). The strong conjecture gives a precise recipe for integers N(ρ) and k(ρ) such that ρ comes from S_{k(ρ)}(N(ρ)). In any particular instance, at least in theory the strong conjecture is verifiable because the space S_{k(ρ)}(Γ_{1}(N(ρ))) is can be computed. In the context of the Shimura-Taniyama conjecture, the difference between the weak and strong conjectures manifests itself in the difference between the assertions “A comes from a modular form somewhere” and “A comes from a modular form of level equal to the conductor of A.”

The level and weight satisfy the following:

1. ℓ ∤ N(ρ), and N(ρ) = ∏_{p≠ℓ} p^{n(p)}, where n(p) depends only on ρ|_{I_{p}}.

2. k(ρ) depends only on ρ|_{I_{ℓ}}.

The integer n(p) is a conductor in additive notation, in particular

n(p) = 0 ⟺ ρ is unramified at p.

For example, suppose ρ : G → GL(2, \overline{F}_{ℓ}) is ramified only at ℓ. The strong conjecture implies that ρ comes from S_{k(ρ)}(Γ_{1}(1)). A form of level 1 gives rise to a representation ramified only at ℓ; the above assertion is the converse. When ℓ = 2, 3 it is known that there are no such representations ρ, so the conjecture holds for these values of ℓ. Quite recently, S. Brueggeman considered the case ℓ = 5; she proved that the conjected result follows from the Generalized Riemann Hypothesis [4]. The case ℓ = 7 still remains wide open.

Now we say more about n(p), the exponent of the conductor at p, but avoid giving a formula. View the representation ρ as ρ : G → Aut(V) where V is a two dimensional vector space. It is natural to consider the subspace

V^{I_{p}} = {v ∈ V : ρ(σ)v = v, all σ ∈ I_{p}}.

For example, V^{I_{p}} = V iff ρ is unramified at p. As a first approximation we have

n(p) = dim(V/V^{I_{p}}) + (wild term).

We now recall the notion of newform. Associated to an eigenform f ∈ S_{k}(Γ_{1}(N)) we have a package (… a_{p} …), p ∤ N, of eigenvalues. We say that f is a newform if the system of eigenvalues comes from no level M | N with M ≠ N. Newforms were studied in [1, 26, 7, 30]. The idea is to understand where the systems of eigenvalues first arise, and see how the full space of forms can be reconstructed from the newforms.

Suppose ρ comes from a newform f ∈ S_{k}(Γ_{1}(N)), but do not assume that k, N are Serre’s optimal k(ρ), N(ρ). What is the difference between N and N(ρ)? Let O be the ring of integers of the field generated by the Fourier coefficients of f and let φ : O → \overline{F}_{ℓ} be the map such that φ(a_{p}) = tr(ρ(Frob_{p})). Let λ be a prime of O lying over ℓ. Deligne constructed a λ-adic representation ρ_{λ} which gives rise to ρ. Set E = Frac(O) and view ρ_{λ} as giving a representation

ρ_{λ} : G ⟶ GL(2, E_{λ}) = Aut(Ṽ)

where Ṽ is a 2 dimensional vector space over E_{λ}. The following diagram summarizes this information.

\xymatrix{ &  & GL(2, E_{λ}) = Aut(Ṽ) \\ G \ar[urr] \ar[drr]_{ρ} \ar[r]^{ρ_{λ}} & GL(2, O_{λ}) \ar[dr]^{φ} \ar@{^{(}->}[ur] \\  &  & GL(2, \overline{F}_{ℓ})}

We wish to compare Ṽ with V. This is like comparing apples with oranges, since one is a vector space over the ℓ-adic field E_{λ} and the other over \overline{F}_{ℓ}. We compare the dimensions of the inertia invariants:

dim Ṽ^{I_{p}} ≤ dim V^{I_{p}}.

Miraculously, it could be that Ṽ is so ramified that Ṽ^{I_{p}} = 0, but V is unramified so dim V^{I_{p}} = 0. The following defines n(p):

ord_{p}(N) = n(p) + (dim V^{I_{p}} − dim Ṽ^{I_{p}}).

Though this may be a convulated way of presenting n(p), it is what is used when trying to prove one of the main theorems in the subject: If ρ is modular at all, it is possible to “refine” N and k and eventually discover that ρ does come from S_{k(ρ)}(Γ_{1}(N(ρ))). After much work it turns out that [for ℓ > 2?]

Weak Conjecture ⟺ Strong Conjecture

Starting with a level N giving ρ we have

n(p) = ord_{p}(N) − (dim V^{I_{p}} − dim Ṽ^{I_{p}}).

Thus we can read of N(ρ) from ord_{p}(N), dim(V^{I_{p}}), and dim(Ṽ^{I_{p}}). In particular, N(ρ) | N for any N giving f. If f ∈ S_{k}(Γ_{1}(N)) and ℓ ∤ N, then k ≤ k(ρ). However, if we allow powers of ℓ in the level then the weight k can always be made equal to 2. Amazingly (to me), Serre’s conjectures imply that all odd irreducible two-dimensional mod ℓ representations of Gal(\overline{Q}/Q) may be realized in spaces of ℓ-division points on such abelian varieties. The weak Serre conjecture states that all such representations come from modular forms, and then it takes only a bit of technique to show that one can take the modular forms to have weight two (if one allows powers of ℓ in the level).

We now consider the representations arising from

B : y^{2} + y = x^{3} + x^{2} − 12x + 2.

The conductor of B is N = 141 = 3 · 47 and there is a newform f ∈ S_{2}(Γ_{0}(141)) attached to B. The curve B is isolated in its isogeny class, so by Exercise 10.3, for every ℓ the representation ρ_{ℓ} : G → GL(2, F_{ℓ}) ≈ Aut(B[ℓ]) is irreducible. Serre also proved that each ρ_{ℓ} is surjective. The (minimal) discriminant of B is Δ = 3^{7} · 47. What are the Serre level and weight or the ρ_{ℓ} attached to B? After a possibly quadratic extension, B becomes a “Tate curve.” This can be used [41] to show that for p ≠ ℓ,

ρ_{ℓ} is unramified at p ⟺ ord_{p}(Δ) ≡ 0 (mod ℓ).

Take p = 3, ℓ ≠ 3: then ρ_{ℓ} is unramified at 3 iff ℓ | ord_{3}(Δ) = 7, i.e., when ℓ = 7. Furthermore, ρ_{ℓ} is always ramified at 47. What does this say about the optimal level, in the sense of Serre’s conjecture? Take ℓ ≠ 3, 47. Then N(ρ_{ℓ}) | N = 3 · 47. If ℓ ≠ 7 then N(ρ_{ℓ}) = N = 3 · 47, and the strong case is verified. If ℓ = 7 then N(ρ_{ℓ}) = 47, and k(ρ_{ℓ}) = 2 since ℓ ∤ 3 · 47.

The remaining cases are ℓ = 3, 47. If ℓ = 47 then N(ρ_{ℓ}) = 3. It is hard to describe k(ρ) axiomatically. Because the cyclotomic character has order ℓ − 1, k(ρ_{ℓ}) ≡ 2 (mod 47 − 1). It turns out that k(ρ_{ℓ}) = 2 + (47 − 1) = 48. In order to verify the strong conjecture we need a mechanism for going from eigenforms in S_{2}(Γ_{0}(141); F_{ℓ}) to eigenforms in S_{48}(Γ_{0}(3)). In Serre’s [2][fix reference] article he constructs such a map for level 1. Koike and others generalized this map to higher level.

Let us now return to the example ℓ = 7, in which N(ρ_{ℓ}) = 47 properly divides N = 3 · 47. The strong conjecture predicts the existence of g ∈ S_{2}(Γ_{0}(47)) giving rise to ρ. Our initial instinct is to look for an elilptic curve A of conductor 47 for which A[ℓ] = B[ℓ]. We look and are amazed to find that there is no elliptic curve of conductor 47! In fact, S_{2}(Γ_{0}(47)) is 4 dimensional, spanned by the 4 Galois conjugates of a single eigenform f = ∑ c_{n}q^{n} whose Fourier coefficients c_{n} generate the full ring of integers in the field K generated by a root of

h(x) = x^{4} − x^{3} − 5x^{2} + 5x − 1.

The discriminant of K is 1957 = 19 · 103 and the eigenvalue c_{2} satisfies h(c_{2}) = 0. Since h(x) ≡ (x + 2)(x^{3} + 4x^{2} + x + 3) (mod 7), there is a prime λ lying over 7 such that O/λ ≅ F_{7}, and the isomorphism sends c_{2} to −2 (mod 7). Counting, we find that #B̃(F_{2}) = 5 so a_{2} = 3 − 5 = −2 = φ(c_{2}), as expected. More generally, for p ∤ 7 · 141, we have φ(c_{p}) ≡ a_{p} mod 7.

“Level optimization:” We are trying to strip 3 from the level. The representation ρ starts at level 141 = 3 · 47 and is unramfied at 3. We try to verify that there is a form of level 47 giving rise to ρ. In general, the easier case, due to Mazur, is when p ≢ 1 mod ℓ. The hard case, due to Ribet, is when p ≡ 1 (mod ℓ).

LECTURE 3

The weight in Serre’s conjecture

Let ρ : G ⟶ GL(2, \overline{F}_{ℓ}) be an irreducible modular Galois representation.

Conjecture. ρ is modular of level N(ρ) and weight k(ρ).

In the last lecture we discussed the level. Now we discuss the weight. The main reference is [16]. We assume that ρ arises from an eigenform of level N = N(ρ) and weight k ≥ k(ρ) and show that k can be “pushed down” to k(ρ).

We have not yet defined k(ρ). In order to find the definition, we must think about forms of low weight and their associated Galois representations. Let f = ∑ a_{n}q^{n} be an eigenform of level N with (N, ℓ) = 1, such that the weight k of f satisfies 2 ≤ k ≤ ℓ + 1, and let ε denote the character of f. By convention, in this lecture we will not consider weight 1 forms, though it is important to think about them in certain context. Let O be the integer ring of Q(… a_{n} …) and fix φ : O → \overline{F}_{ℓ}. We will sometimes cut down on notation by writing a_{n} for φ(a_{n}), i.e., by thinking of a_{n} as in \overline{F}_{ℓ}. Let ρ = ρ_{f}. In the previous lecture we considered ρ|_{D_{p}} for p ≠ ℓ. To understand the weight we instead consider ρ|_{D_{ℓ}}, which is described in many cases by theorems of Fontaine and Deligne. There are two cases to consider:

• ordinary case: a_{ℓ} ≠ 0 ∈ \overline{F}_{ℓ}

• supersingular case: a_{ℓ} = 0 ∈ \overline{F}_{ℓ}

Our goal is to read off the weight from properties of ρ|_{I_{ℓ}}.

Ordinary case. We first consider the ordinary case, due to Deligne. In this case, ρ has a one dimensional unramified quotient, so

ρ|_{D_{ℓ}} ∼ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

with β unramified, i.e., β(I_{ℓ}) = 1 and αβ = χ^{k−1}ε. The character ε is also unramified at ℓ because it is a character of conductor N and N is coprime to ℓ. Thus

ρ|_{I_{ℓ}} ∼ (\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix})

Since χ has order ℓ − 1 we can recover k modulo ℓ − 1. This is enough to determine k when k ≠ 2, ℓ + 1. In Lecture 5 we will see how to decide between 2 and ℓ + 1.

Supersingular case. Fontaine investigated the supersingular case. In order to describe ρ|_{I_{ℓ}} we introduce the fundamental characters of tame inertia. Fix an algebraic closure \overline{Q}_{ℓ} of the field Q_{ℓ} of ℓ-adic numbers. Let Q_{ℓ}^{nr} be the maximal unramified extension and Q_{ℓ}^{tame} the maximal tame extension. The tame extension Q_{ℓ}^{tame} is built up by all finite extensions of Q_{ℓ}^{nr} of degree coprime to ℓ. We have the following tower of fields.

\overline{Q}_{ℓ}_{D_{ℓ}}  _{I_{ℓ}}_{I_{w}} Q_{ℓ}^{tame}_{I_{t}} Q_{ℓ}^{nr} Ẑ Q_{ℓ}

There are many fields intermediate between Q_{ℓ}^{nr} and Q_{ℓ}^{tame}. If n ≥ 1 and ℓ ∤ n then Q_{ℓ}^{nr}( \sqrt[n]{ℓ}) ⊂ Q_{ℓ}^{tame}, and furthermore these extensions generate Q_{ℓ}^{tame}, see [39]. Since ℓ ∤ n, Q_{ℓ} is unramified at primes dividing n and so μ_{n} ⊂ Q_{ℓ}^{nr}. By Kummer theory we understand the Galois group of Q_{ℓ}^{tame}. In particular, there is a canonical isomorphism

Gal(Q_{ℓ}^{nr}( \sqrt[n]{ℓ})/Q_{ℓ}^{nr}) ⟶ μ_{n}, σ ↦ \frac{σ( \sqrt[n]{ℓ})}{\sqrt[n]{ℓ}}

This isomorphism lifts to a map I → μ_{n} which factors through I_{t}. The group μ_{n} sits inside the ring of integers O of \overline{Q}_{ℓ}. Reducing modulo the maximal ideal gives a character

I ⟶ μ_{n}(\overline{Q}_{ℓ}) ⟶^{∼} μ_{n}(\overline{F}_{ℓ}) ⊂ \overline{F}_{ℓ}^{∗}.

Let n = ℓ^{ν} − 1. The surjective map I → F_{ℓ^{ν}}^{∗} is called a fundamental character of level ν. Though this may seem pendantic, think of this \overline{F}_{ℓ} differently than the one at the beginning of the lecture. There are ν different fundamental characters of level ν as illustrated by the following diagram:

\xymatrix{I \ar[dr] &  & \overline{F}_{ℓ} \\  & F_{ℓ^{ν}}^{∗} ⊂ F_{ℓ^{ν}} \ar[ur] \ar[ur] \ar[ur]_{ν maps}}

These characters, taken as a whole, are the fundamental characters of level ν. The fundamental character of level 1 is the mod ℓ cyclotomic character, see Exercise 10.14. When ν = 2, there are two fundamental characters which we denote Ψ and Ψ′. They satisfy Ψ^{ℓ} = Ψ′ and (Ψ′)^{ℓ} = Ψ.

If B/Q_{ℓ} is a supersingular elliptic curve, then since I_{t} is abelian of order prime to ℓ, the representation I_{t} → Aut(B[ℓ]) ⊂ GL(2, \overline{F}_{ℓ}) is diagonalizable, equal to a direct sum of two characters. Amazingly, the two characters are the fundamental characters Ψ and Ψ′. In [36] Serre explains very carefully how we obtain these two characters by studying formal groups. More generally, Fontaine proved that if f is supersingular of weight k ≤ ℓ then ρ_{f}|_{I} : I ⟶ GL(2, \overline{F}_{ℓ}) factors through I_{t} and is a direct sum of the two character Ψ^{k−1} and (Ψ′)^{k−1}.

We now know how to see the weight of a Galois representation arising from a form of low weight. Next start with a representation ρ : D → GL(2, \overline{F}_{ℓ}) which is known to be modular, but arising from a form of some possibly quite large weight k. Let ρ^{ss} denote the simisimplification of ρ, thus ρ^{ss} = ρ if ρ is irreducible, otherwise ρ^{ss} is a direct sum of two characters α and β.

Lemma 3.1. ρ^{ss} is tame, i.e., ρ^{ss}(I_{w}) = 0.

Proof. The wild ramification group I_{w} is the kernel of the map Gal(\overline{Q}_{ℓ}/Q_{ℓ}) → Gal(\overline{Q}_{ℓ}/Q_{ℓ}^{tame}), hence it is a normal subgroup of D. Each finite Galois extension of Q_{ℓ}^{tame} is of degree a power of ℓ, so I_{w} is the profinite Sylow ℓ-subgroup of I.

To simplify notation, write σ = ρ^{ss}. Let W = F_{ℓ^{ν}} × F_{ℓ^{ν}} be the representation space of σ. Then W^{I_{w}} = {w ∈ W : σ(τ)w = w for all τ ∈ I_{w}} is a subspace of W invariant under the action of D. To see this, let α ∈ D, τ ∈ I_{w}, and w ∈ W^{I_{w}}. Since I_{w} is normal in D, α^{−1}τα = τ′ for some τ′ ∈ I_{w}. Therefore

σ(α)^{−1}σ(τ)σ(α)w = σ(τ′)w = w

so σ(τ)σ(α)w = σ(α)w, hence σ(α)w ∈ W^{I_{w}}.

Claim: W^{I_{w}} ≠ 0. Write W as a disjoint union of its orbits under the action of I_{w}. Since I_{w} is an Sylow ℓ-subgroup and W is finite we see that the size of each orbit is either 1, or a positive power of ℓ. Now {0} is a singleton orbit, W has ℓ-power order, and all non-singleton orbits have order a positive power of ℓ, so there must be at least ℓ − 1 other singleton orbits. Each of these gives a nonzero element of W^{I_{w}}.

If W^{I_{w}} = W then I_{w} acts trivially and we are done. If W^{I_{w}} ≠ W, then W^{I_{w}} is nonzero so it is a one dimensional subspace invariant under D. Because σ is semisimple it is diagonal. If τ ∈ I_{w}, then τ has order ℓ^{n} for some n so σ(τ) = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) , with α^{ℓ^{n}} = β^{ℓ^{n}} = 1. But α, β ∈ F_{ℓ^{ν}} so they have order dividing |F_{ℓ^{ν}}^{∗} | = ℓ^{ν} − 1. Since gcd(ℓ^{ν} − 1, ℓ^{n}) = 1 it follows that α = β = 1, so I_{w} acts trivially.

Thus σ = ρ^{ss}|_{I} factors through the (prime-to-ℓ) tame inertia group I_{t}, so σ is diagonalizable, equal to the direct sum of two characters α, β : I → \overline{F}_{ℓ}^{∗}. The characters α, β have some stability properties because σ = ρ^{ss}|_{I} is the restriction of a representation of the full decomposition group. Consider the tower of fields

K( \sqrt[n]{ℓ})
K = Q_{ℓ}^{ur}

Q_{ℓ}

Let G = Gal(K( \sqrt[n]{ℓ})/Q_{ℓ}). Recall that Gal(K( \sqrt[n]{ℓ})/K) ≅ μ_{n}(K) and Gal(K/Q_{ℓ}) is topologically generated by Frob_{ℓ}. If h ∈ μ_{n}(K) and g ∈ G restricts to Frob_{ℓ}, then we have the conjugation formula: ghg^{−1} = h^{ℓ}. Applying this to σ with h ∈ I_{t} we find that

σ(ghg^{−1}) = σ(h^{ℓ}) = σ(h)^{ℓ}

so

σ(g)σ(h)σ(g^{−1}) = σ(ghg^{−1}) = σ(h)^{ℓ}.

The representation h ↦ σ(h)^{ℓ} is equivalent to h ↦ σ(h) via conjugation by σ(g). As a consequence, the pair of characters {α, β} is stable under ℓ-th powering, i.e., as a set {α, β} = {α^{ℓ}, β^{ℓ}}. There are 2 possibilities:

• ordinary case: α^{ℓ} = α, β^{ℓ} = β, so α, β : I → F_{ℓ}^{∗}.

• supersingular case: α^{ℓ} = β ≠ α, β^{ℓ} = α ≠ β, so that α^{ℓ^{2}} = α and β^{ℓ^{2}} = β, and α, β : I → F_{ℓ^{2}}^{∗}.

We now discuss the supersingular case further, leaving the ordinary case for next time. Let

α, β : I ⟶ F_{ℓ^{2}}^{∗}

be the two tame characters associated to the semisimplification σ of ρ|_{I}. Letting Ψ denote one of the fundamental characters of level 2, we write α = Ψ^{n}, β = Ψ^{nℓ}, with n an integer modulo ℓ^{2} − 1. Write n in base ℓ as n = a + ℓb with 0 ≤ a, b ≤ ℓ − 1. Then ℓn ≡ b + ℓa (mod ℓ^{2} − 1). Switching α, β permutes a, b so, relabeling if necessary, we may assume that a ≤ b. If a = b, then α = Ψ^{a}(Ψ′)^{a} = χ^{a}, so α takes values in F_{ℓ}^{∗}, contrary to our dichotomy. Thus we may assume that

0 ≤ a < b ≤ ℓ − 1.

We can factor out a power of the cyclomotic character:

α = Ψ^{n} = Ψ^{a}(Ψ′)^{b} = Ψ^{a}(Ψ′)^{a}(Ψ′)^{b−a} = χ^{a}(Ψ′)^{b−a}
β = χ^{a}Ψ^{b−a}.

Thus

σ ∼ χ^{a} ⊗ (\begin{matrix}Ψ^{b−a} & 0 \\ 0 & (Ψ′)^{b−a} \end{matrix}) .

Setting k = b − a + 1, the untwisted representation is ( \begin{matrix}Ψ^{k−1} & 0 \\ 0 & (Ψ′)^{k−1} \end{matrix} ) . Since 2 ≤ b − a + 1 ≤ ℓ, the weight of the untwisted representation is in the correct range.

What happens to the weight as we start twisting? The answer is a whole chapter in the theory of modular forms. The theory was first developed by Serre and Swinnerton–Dyer, then later jazzed up by Katz [23]. Katz defined spaces of modular forms mod ℓ (see Appendix 9) and a q-expansion map

α : ⨁_{k≥0} M_{k}(Γ_{1}(N); F_{ℓ}) ⟶ F_{ℓ}[[q]].

This map is not be injective, for example, the Hasse invariant A of weight ℓ − 1 has q-expansion 1, and so does the weight 0 form 1, so A − 1 ker(α).

Define an operator θ = q \frac{d}{dq} on q-expansions by θ(∑ a_{n}q^{n}) = ∑ na_{n}q^{n}. In particulary, if ∑ a_{n}q^{n} ∈ F_{ℓ}[[q]] is the q-expansion of a mod ℓ modular form, then a_{p}(θf) = pa_{p}(f). Serre and Swinnerton-Dyer showed that if f is an eigenform of weight k, then there is a mod ℓ eigenform θf of weight k + ℓ + 1 whose q-expansion is θ(∑ a_{n}q^{n}). In particular, when applying θ, the level N does not change.

Define the minimal weight filtration w as follows. If f is a mod ℓ form, let w(f) be the minimal possible k so that the q-expansion of f comes from a modular form of weight k; if no such k exists, do not define w(f). We have the following amazing theorem.

Theorem 3.2. w(θf) = w(f) + ℓ + 1 ⟺ ℓ ∤ w(f).

We can now give the recipe for k(ρ) in the supersingular case. The minimal weight before twisting is b − a + 1 > 0, which is not divisible by ℓ. Each twist adds ℓ + 1 to the weight. Thus in the supersingular case

k(ρ) = (1 + b − a) + a(ℓ + 1) = 1 + ℓa + b.

The point is that the minimal weight does not drop when twisting a times. This can be seen as follows. Since 1 ≤ 1 + b − a < ℓ and (1 + b − a) + a(ℓ + 1) ≤ ℓ^{2}, the weight can only drop if there exists c with 1 ≤ c < a such that

(1 + b − a) + c(ℓ + 1) ≡ 0 (mod ℓ).

If this happens, then c ≡ a − b − 1 (mod ℓ). But 1 ≤ c < a ≤ ℓ, so either c = a − b − 1 < 0, since a < b, or c = ℓ + a − b − 1 = a + ℓ − 1 − b ≥ a, neither of which can happen.

LECTURE 4 Edixhoven’s proof

In this lecture we sketch Edixhoven’s proof that if ρ is modular of type N, k with ℓ ∤ N and ρ is supersingular, then ρ is modular of type N, k(ρ).

Let ρ : G ⟶ GL(2, \overline{F}_{ℓ}) be a supersingular mod ℓ Galois representation, i.e., tr ρ(Frob_{ℓ}) = 0. Let D and I denote the decomposition and inertia groups at ℓ, respectively, and let σ be the semisimplification of ρ|_{D}. We have

σ|_{I} ∼ (\begin{matrix}Ψ^{n} & 0 \\ 0 & (Ψ′)^{n} \end{matrix})

where Ψ and Ψ′ = Ψ^{ℓ} are the two fundamental characters of level 2. Switching Ψ, Ψ′, if necessary, we have

n = a + bℓ, 0 ≤ a < b ≤ ℓ − 1.

Then

Ψ^{n} = Ψ^{a+bℓ} = Ψ^{a}(Ψ′)^{b} = Ψ^{a}(Ψ′)^{a}(Ψ′)^{b−a} = χ^{a}(Ψ′)^{b−a}

so

σ|_{I} ∼ χ^{a} (\begin{matrix}Ψ^{b−a} & 0 \\ 0 & (Ψ′)^{b−a} \end{matrix}) .

Motivated by Fontaine’s local analysis of Galois representations arising from supersingular eigenforms we defined

k(ρ) = a(ℓ + 1) + (b − a + 1) = 1 + ℓa + b.

We explain the ingredients of Edixhoven’s proof in this case. We want to show that if ρ is modular, of level N prime to ℓ, then ρ is modular of weight k(ρ) and level N. The first step is to prove a fairly well-known result, namely that, up to twist, all systems of eigenvalues occur in weight ≤ ℓ + 1.

Theorem 4.1. Suppose ρ is modular of level N and some weight k, and that ℓ ∤ N. Then some twist ρ ⊗ χ^{−α} is modular of weight ≤ ℓ + 1 and level N.

This is a general theorem, applying in both the ordinary and supersingular case. It was discovered by Serre [38] when N = 1; further work was done by Jochnowitz [22] and Ash-Stevens when ℓ ≠ 2, 3. Two proofs are given in Edixhoven’s article.

We now consider an interesting special case. Suppose ρ : G → GL(2, \overline{F}_{ℓ}) is unramified outside ℓ. Then Serre’s conjecture implies that ρ has a twist (by a power of χ) coming from an eigenform on SL(2, Z) of weight at most ℓ + 1. In particular, suppose that ℓ < 11. Then the spaces S_{k}(SL_{2}(Z)) with k ≤ ℓ + 1 are all 0; as a result they contain no non-zero eigenforms! The conjecture that all ρ are modular (of level 1) thus predicts that there are no representations of the type contemplated if ℓ is 2, 3, 5, 7. In support of this conjecture, the non-existence statement was proved for ℓ = 2 by J. Tate in a May, 1973 letter to Serre [46]. Soon after, Serre treated the case ℓ = 3 by methods similar to those of Tate. (See [46] for a discussion and a reference to a note in Serre’s Oevres. [fix typesetting of “Oe”.]) Quite recently, S. Brueggeman considered the case ℓ = 5; she proved that the conjected result follows from the Generalized Riemann Hypothesis [4]. The case ℓ = 7 remains open.

Very often, it is useful to think about modular forms in terms of group cohomology. Let R^{2} be the natural two dimensional vector space on which Γ_{1}(N) acts. The Eichler-Shimura isomorphism is

S_{k}(Γ_{1}(N)) ⟶^{≅} H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(R^{2}))

where H_{P}^{1} is just the subgroup of the usual group cohomology of classes vanishing at the cusps. One defines in a natural way an action of the Hecke algebra T on the right hand side and checks that the above is an isomorphism of T-modules. Eichler-Shimura is a great “nuts and bolts” approach to modular forms. The forms whose “periods” happen to be integral form a lattice

H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(Z^{2})) ⊂ H_{P}^{1} (Γ_{1}(N), Sym^{k−2}(R^{2})).

Reducing modulo ℓ we obtain H_{P}^{1} (Γ̃_{1}(N), Sym^{k−2}(F_{ℓ}^{2})) where Γ̃_{1}(N) is the image of Γ_{1}(N) in SL_{2}(F_{ℓ}). Serre and Hida observed that for k − 2 ≥ ℓ, the Γ̃_{1}(N) representation Sym^{k−2}(F_{ℓ}^{2}) is a sum of representations arising in Sym^{k′−2}(F_{ℓ}^{2}) for k′ < k.

We return now to the case when ρ is supersingular and modular. By the above theory, there is a form f of weight k ≤ ℓ + 1 such that χ^{−α} ⊗ ρ ∼ ρ_{f}. If k = ℓ + 1, by a theorem of Mazur [17, 2.8], there is a form of weight 2 giving rise to ρ_{f}, so we may assume k ≤ ℓ. If k = 1, multiply by the Hasse invariant, so we may assume 2 ≤ k ≤ ℓ. Our goal is to show that w(θ^{α}f) = k(ρ) by understanding well how the θ operator changes the minimal weight. We have ρ_{f} ⊗ χ^{α} ∼ ( \begin{matrix}Ψ^{n} & 0 \\ 0 & (Ψ′)^{n} \end{matrix} ) with n = a + bℓ, a < b. By Fontaine’s theorem (see Lecture 3) the characters corresponding to ρ_{f} are Ψ^{k−1} and (Ψ′)^{k−1}. Thus

{Ψ^{k−1}χ^{α}, (Ψ′)^{k−1}χ^{α}} = {Ψ^{n}, (Ψ′)^{n}}.

We now compute w(θ^{α}f), assuming for simplicity that ℓ is odd. There are two cases to consider.

Case 1. First suppose

Ψ^{k−1}χ^{α} = Ψ^{k−1+α(ℓ+1)} = (Ψ′)^{n} = Ψ^{b+aℓ}.

Then k − 1 + α(ℓ + 1) ≡ b + aℓ (mod ℓ^{2} − 1) so k − 1 ≡ b − a (mod ℓ + 1) and hence k = 1 + b − a. Next, reduce modulo ℓ − 1 and substitute k = 1 + b − a giving b − a + 2α ≡ b + a (mod ℓ − 1) so α = a (note, α is only defined modulo ℓ − 1.) Applying Theorem 3.2 and arguing as in the end of Lecture 3 shows that

w(θ^{a}f) = w(f) + a(ℓ + 1) = 1 + b − a + aℓ + a = 1 + b + aℓ = k(ρ).

Case 2. The second possibility is that

Ψ^{k−1}χ^{α} = Ψ^{k−1+α(ℓ+1)} = Ψ^{n} = Ψ^{a+bℓ}.

Then k − 1 ≡ a − b (mod ℓ + 1) so

k = ℓ + 2 − (b − a).

If b−a = 1, then k = ℓ+1, contrary to our assumption that 2 ≤ k ≤ ℓ, so b−a > 1. We have

k − 1 + α(ℓ + 1) ≡ a + bℓ (mod ℓ − 1)

so that, upon substituting k = ℓ + 2 − (b − a) we find that α = b − 1. We must now verify that w(θ^{b−1}f) =^{?} w(ρ) = 1 + b + aℓ. Unfortunately k = ℓ + 2 − (b − a) is not especially telling, in particular it is not possible to apply the argument of case 1 to compute w(θ^{α}f).

Because f is supersingular, Fermat’s Little Theorem implies that θ^{ℓ−1}f = f. To compute w(θ^{b−1}f) we use Tate’s theory of θ-cycles, which was written down in [22] and recalled in [17]. The θ-cycle for a supersingular eigenform must exhibit the following pattern:

[figure: •; dip once; •; final dip; •; go up; •; go up; •]

go up, dip once, go up, final dip back to original weight

From this we can deduce the θ-cycle. Starting with k we list ℓ numbers starting and ending with k.

\begin{matrix}k, k + (ℓ + 1), k + 2(ℓ + 1), …, k + (ℓ − k)(ℓ + 1), \\ ℓ + 3 − k, (ℓ + 3 − k) + (ℓ + 1), …, (ℓ + 3 − k) + (k − 3)(ℓ + 1), \\ k \end{matrix}

The first and second lines contain ℓ + 1 − k and k − 2 numbers, respectively. All told, we have listed ℓ numbers. Now we can compute w(θ^{b−1}f). If

b − 1 ≤ ℓ − k = ℓ − (ℓ + 2 − b + a) = −2 + b − a

then a ≤ −1, a contradiction. Thus we may assume that b − 1 > ℓ − k so

w(θ^{b−1}f) = ℓ + 3 − k + (ℓ + 1)(b − 2 − (ℓ − k)) = 1 + b + aℓ = k(ρ)

which verifies Serre’s conjecture in this case.

Ordinary case. Finally we briefly consider the ordinary case, in which

ρ|_{I} ∼ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

with α, β : I → F_{ℓ}^{∗} both powers of the cyclotomic character. We can view this as a twist of a representation where the lower right entry is 1:

(\begin{matrix}α & ∗ \\ 0 & β \end{matrix}) ∼ β ⊗ (\begin{matrix}αβ^{−1} & ∗ \\ 0 & 1 \end{matrix}) .

We have to develop an ordinary version of θ-cycles. In general this can be complicated. Assume that β = 1 to simplify the situation, so the representation looks like

(\begin{matrix}χ^{i} & ∗ \\ 0 & 1 \end{matrix}) with 1 ≤ i ≤ ℓ − 1. Recall that if f is of weight k, then the associated

representation is ( \begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix} ) with 2 ≤ k ≤ ℓ + 1. Our initial reaction is to define k(ρ) to be i + 1. But the extreme weights 2 and ℓ + 1 are “the same.” If we start with a form of weight either 2 or ℓ + 1 and look at the corresponding representation, this recipe would predict that k(ρ) = 2, which is wrong in general. For example, suppose f = Δ is the level 1 cusp form of weight 12 and ρ = ρ_{f}. If we try to make the weight of ρ equal to 2, then we fall flat! There is no such form.

LECTURE 5

Galois representations from modular forms

Suppose ρ : Gal(\overline{Q}/Q) → GL(2, F_{ℓ}) is ordinary. In the simplest case, ρ|_{I} ∼ ( \begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix} ) with 2 ≤ k ≤ ℓ + 1. The weight k(ρ) ≡ k (mod ℓ − 1); it remains to choose between 2 and ℓ + 1. The key is “good reduction.” Let f = ∑ a_{n}q^{n} be a normalized eigenform of weight 2. We try to give the flavor of the extra property that ρ = ρ_{f} will have. Let O be the ring of integers of the totally real or CM field E = Q(… a_{n} …). We associate to f an abelian variety A/Q.

[figure: eigenforms f; abelian varieties A/Q]

If all a_{n} lie in Q then dim A = 1. More generally, A is an abelian variety of dimension equal to the degree [E : Q]. Furthermore, A is furnished with an action of O:

O ↪ End_{Q} A.

Where do we find A? Bibliographically speaking, we can find A in Shimura’s book [44, Theorem 7.14]. But where, in an arithmetic sense? If f has level N, associated to N there is a modular curve X_{1}(N)/Q

X_{1}(N) = \overline{Γ_{1}(N)\𝔥} = Γ_{1}(N)\𝔥 ∪ \{cusps\}

where 𝔥 is the open upper half plane. The Jacobian

J_{1}(N)/Q = Jac(X_{1}(N))

is an abelian variety over Q of dimension equal to the genus g of X_{1}(N), and having good reduction at all primes ℓ ∤ N. For example, when N = 1, we have X_{1}(1) ≈ ¶^{1}, and J_{1}(1) = 0. The abelian variety A associated to F is constructed as a quotient of J_{1}(N). In particular, the reduction of A at each prime ℓ ∤ N is good.

The representation ρ = ρ_{f} is found inside of A. More precisely, let λ = ker(O → \overline{F}_{ℓ}) and set

A[λ] = \{P ∈ A(\overline{Q}) : xP = 0 all x ∈ λ\}.

Since O is the full ring of integers, dim_{O/λ} A[λ] = 2. The Galois representation is then

\xymatrix{G \ar[rrr]^{ρ} \ar[dr] &  &  & GL(2, \overline{F}_{ℓ}) \\  & Aut_{O/λ} A[λ] \ar[r]^{≈} & GL(2, O/λ) \ar@{^{(}->}[ur]}

Let I = Gal(\overline{Q}_{ℓ}/Q_{ℓ}^{nr}) be the inertia group at ℓ. We say that the finite group scheme A[λ] has good reduction if it extends to a finite flat group scheme over O_{Q_{ℓ}^{nr}}. This is the case for A[λ]; the finite flat group scheme extending A[λ] is the scheme theoretic closure of A[λ] in a good model A/O_{Q_{ℓ}^{nr}} of A, which exists because A has good reduction at ℓ. In general, if ρ is a representation and k(ρ) ≡ 2 (mod ℓ − 1), then

k(ρ) = 2 ⟺ ρ comes from finite flat group scheme /O_{Q_{ℓ}^{nr}}

A typical example. Let A/Q be an elliptic curve with discriminant Δ_{A} and conductor N_{A}. Suppose that ℓ^{2} ∤ Δ_{A} so that A has semistable reduction at ℓ. Consider the representation ρ = ρ_{A,ℓ} coming from the Galois action on A[ℓ]. Make the hypothesis that A[ℓ] is irreducible.

Theorem 5.1. Criterion for ρ to be finite flat at ℓ:

k(ρ) = 2 ⟺ ℓ | ord_{ℓ}(Δ_{A})

Criterion for ρ to be unramified at ℓ: For p ≠ ℓ,

ρ unramified at p ⟺ ℓ | ord_{p} Δ_{A}.

Frey curves. For example, we can apply these criteria to the Frey curves associated to hypothetical solutions of the Fermat equation a^{ℓ} + b^{ℓ} = c^{ℓ}, with ℓ ≥ 11. Let A be the elliptic curve given by the equation y^{2} = x(x − a^{ℓ})(x + b^{ℓ}). Then

Δ_{A} = \frac{((abc)^{2})^{ℓ}}{2^{8}} .

By a theorem of Mazur’s, A[ℓ] is irreducible and, applying the above theorem, the associated representation ρ is unramified away from the prime 2 and ℓ. Thus N(ρ) | 2, and k(ρ) = 2 since ℓ | ord_{ℓ}(Δ_{A}).

Elliptic curves of conductor 33. There is a unique normalized newform f = ∑ a_{n}q^{n} on Γ_{0}(33) so a_{n} ∈ Q for all n. There is one isogeny class of elliptic curves of conductor 33; let A be the one given by y^{2} + xy = x^{3} + x^{2} − 11x. Then Δ_{A} = 3^{6} · 11^{2}. The mod ℓ = 3 representation ρ : G → Aut(A[3]) is surjective. Since 3 | ord_{3} Δ_{A} = 6, k(ρ) = 2 and N(ρ) = 11. Sure enough, there is modular form with N = 11 and k = 2. Geometrically, choose one of the three elliptic curves of conductor 11 (it doesn’t matter which) and B[3] ≈ A[3] as representations of Gal(\overline{Q}/Q). For example, the eigenforms corresponding to A and B,

f_{A} = q − 2q^{2} − q^{3} + 2q^{4} + q^{5} + 2q^{6} − 2q^{7} − 2q^{9} + ⋯
f_{B} = q + q^{2} − q^{3} − q^{4} + −2q^{5} − q^{6} + 4q^{7} − 3q^{8} + q^{9} + ⋯

have the same a_{p} modulo 3.

Level 141 example. Let A be the elliptic y^{2} + y = x^{3} + x^{2} − 12x + 2 of conductor 141 = 3 · 47 and having discriminant Δ_{A} = 3^{7} · 47 (we considered in Lecture 1). First set ℓ = 7. Then 7 | ord_{3}(Δ), so ρ = ρ_{A,7} has invariants k(ρ) = 2 and N(ρ) = 47. Indeed, we find a form f ∈ S_{2}(Γ_{0}(47)) with coefficients in a quartic field which gives rise to ρ. Next suppose ℓ = 3 | 141 and let ρ = ρ_{A,3}. Then N(ρ) = 47 and 3 ∤ ord_{3}(Δ) so k(ρ) = ℓ + 1 = 4. Using a computer we find that S_{4}(Γ_{0}(47)) contains two conjugacy classes of eigenforms, one defined over a field of degree 3, and the other over a field of degree 8. The degree 3 form gives rise to ρ_{A,3}.

Companion forms.

Suppose f is ordinary of weight k, that 2 ≤ k ≤ ℓ + 1, and that

ρ_{f}|_{I} ∼ (\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix}) .

Questions: Is this split or not, i.e., does ∗ = 0? How often does this split? The split primes are in the minority. How can one quantify the number of split primes? I don’t know.

For now, let us suppose that ∗ = 0. Then

ρ ∼ (\begin{matrix}1 & 0 \\ 0 & χ^{k−1} \end{matrix}) , so ρ ⊗ χ^{ℓ−k} ∼ (\begin{matrix}χ^{ℓ−k} & 0 \\ 0 & 1 \end{matrix}) .

In particular, k(χ^{ℓ−k} ⊗ ρ) = 1 + ℓ − k. This representation is visibly modular, of some weight and level. To say that it is modular of the conjectured weight is to make a very strong statement: If χ^{ℓ−k} ⊗ ρ is indeed modular of weight 1 + ℓ − k then there exists an eigenform g of weight 1 + ℓ − k with ρ_{g} ∼ χ^{ℓ−k} ⊗ ρ_{f}. The form g is called a companion form. The existence of g is completely nonevident. If a companion form g does exist, then the representation must split, unless perhaps if the two characters are equal.

Special cases. Suppose k(ρ) = ℓ, so the companion has weight 1, i.e., if ρ is unramified at ℓ then ρ comes from a weight 1 eigenform. Suppose k(ρ) = ℓ + 1. If ρ is unramified then ρ comes from weight ℓ + 1 − k = 0, which can not happen. Thus if ρ is unramified the minimal weight must be 2.

The existence of a companion form was largely proved (modulo some unchecked compatibilities) by Gross in [20] for k < ℓ, and in a few cases when k = ℓ. Coleman and Voloch [9] finished the remaining cases except k = ℓ = 2, while avoiding the need to check the unchecked compatibilities. Independent work is under way by Coleman and Taylor to obtain the last remaining case.

LECTURE 6

Introduction to level lowering

Let ρ : G → GL(2, \overline{F}_{ℓ}) be an odd irreducible modular Galois representation. Is ρ modular of the optimal weight k(ρ) and level N(ρ)? In the previous lectures we sketched what is known about optimality of the weight. In this lecture, we introduce some of the techniques involved in optimizing the level.

Suppose ρ arises from a form f of level N. We may assume that N is prime to ℓ. This can be accomplished by multiplying f by a suitable Eisenstein series and taking traces (see [34]). We want to “lower” N to the optimal level N(ρ) (the prime-to-ℓ artin conductor of ρ). Provided f is new, the defect N/N(ρ) can be seen concretely in terms of inertia invariants.

We first reduce to the case k ≤ ℓ + 1 by twisting ρ by an appropriate power of the cyclotomic character. Twisting can be viewed as applying the θ = q \frac{d}{dq} operator, so twisting preserves the level, and it even preserves the conjectured minimal level N(ρ). Thus we may as well assume that the form f giving rise to ρ has weight satisfying 2 ≤ k ≤ ℓ + 1. We can also assume, by Edixhoven’s theory [17], that k = k(ρ). In particular, if k(ρ) = 2 then we shouldn’t work with a form of weight ℓ + 1. Since twisting by χ changes the weight by ℓ − 1, the only ambiguity occurs when k = 2 or ℓ + 1. Thus if 2 ≤ k ≤ ℓ + 1, then automatically k = k(ρ) unless k = 2 or ℓ + 1 in which case we can change the weight, if necessary.

As explained above, we make a replacement f ↦ g = f ⊗ χ^{i} and consider the mod ℓ representation ρ arising from g. The level of g may be much larger than N(ρ), but we have insured that the weight satisfies 2 ≤ k ≤ ℓ + 1. How do we proceed? There are two philosophies:

1. Cohomological machinery: Representations of a given weight come from Deligne’s étale cohomology groups. Fearlessly use all available theory. This is the approach used by Faltings, Jordan, and Livne.^{1}

2. Physical division points: Reduce to weight 2 and use that such representations arise from physical division points on Jacobians of modular curves. We are comfortable with these down to earth objects.

In these lectures, we will pursue the second approach. The key general fact, due to Serre, Koike and others, is that mod ℓ forms of level N and weight k satisfying 2 < k ≤ ℓ + 1 correspond to weight 2 forms of level ℓN.

1This approach seems to be “chicken” of weight 2. [[Delete or rephrase.]]

[figure: level N, 2 < k ≤ ℓ + 1; level ℓN, k = 2]

Note that when k = 2, “forms of level N and weight 2 correspond to forms of level N and weight 2.” Thus we can work with weight k = 2 and level

N^{∗} = \begin{cases}N & if k = 2, \\ Nℓ & otherwise. \end{cases}

How can we visualize ρ inside of a Jacobian, such as J = J_{1}(N^{∗})? In order to understand these representations, we must learn how to “push them around” inside these geometric objects. Let

T = Z[… T_{n} …] ⊂ End_{Q} J

be the Hecke algebra. As a Z-module, T is free of finite rank. In certain cases, for example when the level is prime, T is an order in a product of rings of integers of number fields. More generally, T can fail to be semisimple Exercise 2. In any case, T is a commutative Noetherian ring. It is helpful to view a normalized eigenform as a homomorphism

f : T ⟶ O = Z[… a_{n} …] ⊂ C, T_{n} ↦ a_{n}.

Adding to this datum a map φ : O → \overline{F}_{ℓ} gives the connection with \overline{ρ}. Then a_{p} ↦ tr(ρ(Frob_{p})) ∈ \overline{F}_{ℓ}. Composing the homomorphism coming from f with φ gives an exact sequence 0 → 𝔪 → T → \overline{F}_{ℓ} with 𝔪 a maximal ideal. Thus φ embeds T/𝔪 ↪ \overline{F}_{ℓ}.

Having attached a maximal ideal 𝔪 ⊂ T to the representation ρ, we discover an explicit connection with J = J_{1}(N). Consider

J[𝔪] = \{P ∈ J(\overline{Q}) : xP = 0 all x ∈ 𝔪\} ⊂ J(\overline{Q})_{tor}.

Since the endomorphisms in T ⊂ End_{Q}(J) are Q-rational, J[𝔪] is equipped with an action of G = Gal(\overline{Q}/Q). Further, since 𝔪 acts trivially, J[𝔪] is a vector space over the finite field T/𝔪.

What is the smallest field over which ρ has a model? We have tr(ρ(Frob_{p})) ∈ T/𝔪 ↪ \overline{F}_{ℓ} and det(ρ(Frob_{p})) ∈ (T/𝔪)^{∗} ⊂ \overline{F}_{ℓ}^{∗}, so it looks as though ρ has a model over the finite field T/𝔪. Indeed, such a model exists for the rather coarse reason that Br(F_{ℓ}) = 0. Therefore ρ has a model, call it ρ_{𝔪}, over T/𝔪.

Using that ρ_{𝔪} is absolutely irreducible [3, 32] show that always

J[𝔪] ≈ ⨁_{i=1}^{t} ρ_{𝔪}.

In the 1940’s Weil showed that End(J) ⊗ Z_{ℓ} operators faithfully on the Tate module of J at ℓ, so t is strictly positive. When t = 1 we say that ρ_{𝔪} occurs with multiplicity one.

Under what circumstances is multiplicity one known? A detailed summary of multiplicity one results can be found in [17, §9]. Some supplementary results are contained in [48, Ch. 2]. We know of no numerical examples in which multiplicity one fails.

The techniques used to prove multiplicity one results were first pioneered by Mazur [28] who considered J_{0}(N), N prime. For each 𝔪 ⊂ T maximal he looked at ρ_{𝔪} and asked, “Is J[𝔪] ≈ ρ_{𝔪}?” The answer is yes, if ℓ ≠ 2; and yes – sort of – if ℓ = 2 and ρ_{𝔪} is reducible. He did not treat the case ℓ = 2, ρ_{𝔪} ordinary (T_{2} ∉ 𝔪). The paper [29] elaborates on Mazur’s results in various directions, and [20] proves multiplicity one in many cases in the “situation” J_{1}(N^{∗}), N^{∗} = Nℓ.

At present, multiplicity one is known in all cases except perhaps when all of the following stringent hypothesis on ρ = ρ_{𝔪} are satisfied:

• ℓ = k,

• ρ is unramified at ℓ,

• ρ is ordinary at ℓ, i.e., T_{ℓ} ∉ 𝔪, and

• ρ|_{D_{ℓ}} ∼ ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix}) with α = β.

In [20] Gross shows the existence of companion forms when α ≠ β by first proving multiplicity one. In contrast, Coleman-Voloch [9] showed existence of companion forms when α = β (and ℓ > 2) using a method which completely avoided the need to prove multiplicity one.

We now motivate the above stringent hypothesis, when ℓ = 2. Suppose that ℓ = 2, then k is normalized to be between 2 and ℓ + 1 so k = 2, 3. If k = 3 then ρ is badly ramified at 2 which can be used to deduce multiplicity one [[?]] . So assume ρ is unramified at 2 so that k = 2. If ρ is supersingular, so T_{2} ∈ 𝔪, then I_{2} operates by powers of the 2 fundamental characters of level 2. These have order ℓ^{2} − 1 = 3 ≠ 1, so in the supersingular case, ρ is definitely ramified. Thus assume ρ is ordinary. Then ρ|_{D_{2}} ∼ ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix}) with β unramified, β(Frob_{2}) = T_{2} mod 𝔪. Since k = 2, the determinant αβ of ρ|_{D_{2}} is the mod 2 cyclotomic character χ^{k−1} = χ, which amazingly enough is unramified! Thus α is unramified. Since χ(Frob_{2}) = 1 and αβ = χε, we have

α(Frob_{2}) = β^{−1}(Frob_{2})ε(2) = a_{2}^{−1}ε(2) (mod 𝔪)

where ε : (Z/NZ)^{∗} → O^{∗} is the character of f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{1}(N), ε). The further “bad condition” is that α|_{D_{2}} = β|_{D_{2}} which reexpressed in terms of the image of Frobenius becomes a_{2} = a_{2}^{−1}ε(2) ∈ O/𝔪, or equivalently a_{2}^{2} = ε(2) ∈ O/𝔪. [[To what extent does this condition on a_{2} put us in the exceptional case?]]

We have set our problem up so that level lowering pertains to weight 2 forms of appropriate level, and takes place on Jacobians of modular curves. This problem was described and partially treated in a long paper of Carayol [6].

Salient situation. Assume that ρ arises from a weight 2 form f of level N^{∗} with N^{∗} = pM, p ∤ M, and p ≠ ℓ. Moreover, as could easily happen, assume that ε : (Z/N^{∗}Z)^{∗} → O^{∗} has “nothing to do with p” in the sense that ε factors through the natural map to (Z/MZ)^{∗}. In the salient situation f, a priori on Γ_{1}(N), is really “on” the bigger group Γ_{1}(M)∩Γ_{0}(p). The word “on” used in this context is special to the area of modular forms; it means that f satisfies the appropriate invariance with respect to the group. Finally, make the extra hypothesis that p ∤ N(ρ), which, since precisely the ramified primes divide Serre’s minimal level, is the same as hypothesizing that ρ is unramified at p.

Example 6.1 (Salient example). We recall an example from the previous lectures which illustrates the salient situation. The representation ρ = A[7] arises from the 7-division points of a modular elliptic curve A of conductor N_{A} = 3 · 47 and discriminant Δ = 3^{7} · 47. The eigenform f corresponding to A has trivial character ε and is on Γ_{0}(3 · 47). Since ρ is unramified at 3, the Serre level is N(ρ) = 47 so we must position ourselves to get p = 3 out of the level.

Example 6.2 (Fictional example). The elliptic curves Frey associated with hypothetical solutions to the Fermat equation x^{ℓ} + y^{ℓ} = z^{ℓ} give rise to Galois representations which, according to Shimura-Taniyama, fall into the salient situation. Since k = 2, N = N^{∗} = 2L with L (big!) and square free. At the same time, N(ρ) = 2. We take p any prime dividing L and try to lower the level. If we succeed, repeat the process with the residual factor.

In the salient situation there are in fact two cases:

• Hard case: When two conditions are satisfied simultaneously:

1. p ≡ 1 (mod ℓ) (always true if ℓ = 2), and

2. ρ(Frob_{p}) = ( \begin{matrix}α & 0 \\ 0 & α \end{matrix}) is a scalar matrix. This makes sense because p ∤

N(ρ). Also, since α^{2} = χ^{k−1}ε, we know α up to ±1.

• Easy case: The complementary case. Often referred to as “Mazur’s princi-

ple.” Example 6.1 falls into the easy case because 3 ≢ 1 (mod 7).

Before concluding this lecture, we set up some of the commutative algebra that is required in order to lower levels in the easy case of the salient situation. There are two injective maps

\xymatrix{S_{2}(Γ_{1}(M)) \ar@{^{(}->}[r] \ar@{^{(}->}[r] & S_{2}(Γ_{1}(M) ∩ Γ_{0}(p))} .

One is the inclusion f(q) ↦ f(q) and the other is f(q) ↦ f(q^{p}) (Exercise 10.18). Let S_{2}(Γ_{1}(M) ∩ Γ_{0}(p))_{p-new} be the complement in S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)) of the image of S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)) with respect to the Peterson inner product. Let T be the ring of Hecke operators acting on S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)). If p ∤ M, then T_{p} acts as a direct sum of two copies of its action on S_{2}(Γ_{1}(M)), otherwise T_{p} can “permute” the two factors around (Exercise 1). The action of T on

S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)) ⊂ S_{2}(Γ_{1}(M) ∩ Γ_{0}(p))

is through a quotient \overline{T}. Given a representation ρ associated to a maximal ideal 𝔪 of T we ask: is the image of 𝔪 in \overline{T} contained in a maximal ideal, or alternatively, is it the unit ideal? Lowering the level is equivalent to showing that 𝔪 arises by pullback from a maximal ideal of \overline{T} (Exercise 10.22).

LECTURE 7 Mazur’s principle

Let ρ : G → GL(2, \overline{F}_{ℓ}) be a modular representation coming from level N with ℓ ∤ N. The goal is to “lower” N to N(ρ). Without loss, upon applying [17], we may assume that the weight k equals k(ρ) and hence satisfies 2 ≤ k ≤ ℓ + 1. Set

N^{∗} = \begin{cases}N & if k = 2, \\ Nℓ & otherwise. \end{cases}

Then ρ comes from a form f = ∑ a_{n}q^{n} of weight 2 on Γ_{1}(N^{∗}). Let 𝔪 ⊂ T be the maximal ideal associated to ρ; it is the kernel of the map T → \overline{F}_{ℓ} given by T_{p} ↦ \overline{a}_{p}(f) = tr(ρ(Frob_{p})) ∈ \overline{F}_{ℓ} (p ∤ N^{∗}). In particular, ℓ ∈ 𝔪. The representation ρ is closely related to a the subspace V = J[𝔪] ⊂ J[ℓ] of the ℓ-torsion on the Jacobian J = Pic^{o}(X_{1}(N^{∗})). Goal: Supposing p | N^{∗}/N(ρ), the goal is to replace N^{∗} by a divisor of N^{∗}/p.

The full tree of possibilities was discussed by Carayol [6] (?) and Livne [27]. When ℓ ≥ 3, Carayol treated all but one possibility: the salient case. To get a better idea of what Carayol did, suppose p || N^{∗} but p ∤ N(ρ), so ρ is unramified at p. In particular, det(ρ) = χ_{ℓ}^{k−1}\overline{ε} = χ_{ℓ}\overline{ε} is unramifiied at p. Naturally speaking, ε is a map ε : (Z/N^{∗}Z)^{∗} → O^{∗} ↪ C^{∗}. Composing with φ : O → \overline{F}_{ℓ} gives \overline{ε}.

What does it mean for \overline{ε} to be unramified at p? By the Chinese Remainder Theorem, (Z/N^{∗}Z)^{∗} = (Z/pZ)^{∗} × (Z/MZ)^{∗}, with M = N/p. We can thus factor ε as a product of two characters, ε = ε_{p} ⋯ ε^{(p)}. Since the conductor of ε^{(p)} divides M, the second character is unramified at p. On the other hand, ε_{p} is totally ramified at p. Therefore \overline{ε} is unramified at p precisely when \overline{ε}_{p} = 1. Or, what is the same, the image of \overline{ε}_{p} has ℓ-power order. The order of (Z/pZ)^{∗} is p − 1, so if ε is nontrivial, then p ≡ 1 (mod ℓ). Carayol showed that in this unramified situation (except in a few cases when ℓ = 2, 3), one can find another form f giving rise to ρ whose character ε′ is trivial at p: ε′_{p} = 1. [[Is this right?]]

We once again find ourselvs in the salient situation. Carayol has reduced to the case ε_{p} = 1, so that the form f giving rise to ρ is on Γ_{1}(N^{∗}/p) ∩ Γ_{0}(p). We are reduced to the case

1. p || N,

2. the character is trivial at p, and

3. p ∤ N(ρ) (ρ is unramified at p).

The goal is to show that ρ comes from Γ_{1}(M), M = N^{∗}/p. This goal can be achieved in the following cases:

• Mazur’s principle: If either (1) ρ(Frob_{p}) is not a scalar, or (2) p ≢ 1 (mod ℓ). Note that ρ(Frob_{p}) makes sense because ρ is unramified at p.

• Multiplicity one: If when we try to realize ρ inside of J_{1}(N^{∗}) it occurs, on the nose, with multiplicity one:

J[𝔪] = ⨁ ρ_{𝔪} = ρ_{𝔪} (one copy!)

We know that this is the case most of the time, but there are problems when k(ρ) = ℓ and the eigenvalues of Frob_{p} are not distinct.

• ℓ > 2: This was proved in my “motives paper” [34]. A more refined account can be found in [13].

One encounters real problems when trying to push the third argument through when ℓ = 2. However, as was recently pointed out by Kevin Buzzard, the multiplicity one approach does work when ℓ = 2! In the remainder of this lecture we sketch “Mazur’s principle.”

First we describe a priori what local representation ρ|_{G_{p}} of G_{p} = Gal(\overline{Q}_{p}/Q_{p}) looks like in the salient case. Imagine that ρ arises from a form f of weight 2, level Mp with (M, p) = 1, and the character ε of f is a character on (Z/MZ)^{∗}. Then ρ|_{G_{p}} resembles the mod ℓ representation coming from a Tate curve. More precisely, ρ|_{G_{p}} ∼ ( \begin{matrix}αχ & ∗ \\ 0 & α \end{matrix}). Because we are in the salient situation the two characters αχ and α are unramified at p. Further, α(Frob_{p}) = \overline{a_{p}(f)} and αχ(Frob_{p}) = p\overline{a_{p}(f)}. Applying both sides of the identity det(ρ|_{G_{p}}) = α^{2}χ = εχ to Frob_{p} yields ε(p) = \overline{a}_{p}^{2}. For example, on Γ_{0}(M) we have ε = 1 so a_{p}^{2} = 1. So when f really does correspond to an ellipti curve it corresponds to a genuine Tate curve with either split or nonsplit multiplicative reduction at p, depending on the sign of a_{p} (Exercise 10.23). [[Insert remarks about Langland’s excitement about generalizations of this, esp. to ℓ-adic representations of higher weight (Antwerp).]]

The representation ρ comes from an eigenform f. Attached to f is a maximal ideal 𝔪:

0 ⟶ 𝔪 ⟶ T \underrightarrow{{T_{n} ↦ a_{n} \atop ⟨d⟩ ↦ ε(d)}} O ⟶ \overline{F}_{ℓ}.

The determinants and the traces of ρ lie in T/𝔪 ⊂ \overline{F}_{ℓ}. We can form a model for ρ over T/𝔪, by brute force if necessary. Give the vector space affording ρ over T/𝔪 the name V, so we have

ρ_{𝔪} : G_{Q} ⟶ Aut_{T/𝔪} V, and V ≈ (T/𝔪)^{⊕2}.

Next we realize ρ_{𝔪} as a group of division points in a Jacobian. The modular curve X_{1}(Mp) corresponding to Γ_{1}(Mp) covers the curve X_{1}(M, p) corresponding to Γ_{1}(M)∩Γ_{0}(p). The induced map J = Jac(X_{1}(M, p)) → J_{1}(Mp) = Jac(X_{1}(Mp)) has a small, not-so-interesting, finite kernel on which the Galois action is abelian. The Hecke algebra associated to Γ_{1}(M) ∩ Γ_{0}(p), constructed as a ring of correspondence, can viewed as a subgring T ⊂ End_{Q}(J). The action of T on modular forms then comes by thinking of S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)) as the tangent space to the identity of J. Inside our “geometric gadget” J we find the nonzero G_{Q}- and T/𝔪-module J[𝔪] ≈ ⊕_{i=1}^{t}V. For this discussion we do not even need to know that J[𝔪] is a direct sum of copies of V. The following weaker assertion, known long ago to Mazur [28], will suffice: J[𝔪] is a successive extension of copies of V. In particular, V ⊂ J[𝔪].

A weaker conclusion than that which concluded the previous paragraph is that V ⊂ J[ℓ], since ℓ ∈ 𝔪. Our standing hypothesis that ρ is unramified at p translates into the inclusion V ⊂ J[ℓ]^{I_{p}}. Recalling [42], if A is an abelian variety over Q with good reduction at p, then A[ℓ]^{I_{p}} ≅ A_{F_{p}}[ℓ]. However, J = J_{1}(M, p) does not have good reduction at p since p is implicated in the level. The modular curve X_{1}(M, p) has a complicated, in particular bad, reduction at p, so J is likely to (and in fact does) have bad reduction at p.

Enter the Néron model J of J. Passing to the scheme theoretic closure we have, inside of J, a two dimensional T/m-vector space scheme V. We [12, 24] know a great deal about the complicated bad reduction of X_{1}(M, p). The Deligne-Rapoport model is often illustrated by the following fairly standard squiggly diagram:

X_{1}(M, p) X_{1}(M, p)

The reduction X_{1}(M, p)_{F_{p}} is the union of 2 copies of X_{1}(M), intersecting transversely at the supersingular points. (A supersingular point E ∈ X_{1}(M) is a point which represents an enhanced supersingular elliptic curve.) Raynaud showed how to compute the Néron model of J using our knowledge about models of X_{1}(M, p) over Z_{p}.

Recall the “p-old quotient of T.” The subspace S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)) of S_{2}(Γ_{1}(M) ∩ Γ_{0}(p)) is stable under the Hecke algebra T so there is a map T → End(S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M))). The p-old quotient is the image \overline{T}. The image of 𝔪 in \overline{T} is an ideal \overline{𝔪}. The goal in level lowering in the salient case is to show that \overline{𝔪} is not the unit ideal.

One extracts from [12, 31] that there is an exact sequence

0 ⟶ T ⟶ J_{F_{p}}^{0} ⟶ J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}} ⟶ 0 (7.1)

with T a torus (so T_{\overline{F}_{p}} ≈ G_{m} × ⋯ × G_{m}) and J^{0} the identity component of the Néron modem J of J. Furthermore, one gets a precise description of the torus and of the extension. This exact sequence is T-invariant and \overline{T} can also be viewed as coming from the action of T on J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}}.

The next step is to show that V which starts life living inside J in fact lies in J^{0}. It can be shown that the group Φ = J /J^{0} of components is “Eisenstein” so it can be neglected in the study of irreducible representations. Since V is irreducible the component group J /J^{0}, as a Galois module, has nothing to do with V. Thus we have

[figure: 0; T; J_{F_{p}}^{0}; J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}}; 0; V = V_{F_{p}}]

Since 𝔪 kills V, the image of V in J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}} is killed by the image \overline{𝔪} of 𝔪. If \overline{𝔪} = (1), then V must go to 0 in J_{1}(M)_{F_{p}} × J_{1}(M)_{F_{p}}. By exactness, V is contained in the torus: V ⊂ T.

Upon making explicit the action of T on the torus we will discover that

1. Frob_{p} acts as a scalar on V, and

2. p ≡ 1 (mod ℓ). We are in the “Mazur case”, so one of these conditions fails allowing us to conclude that \overline{𝔪} is not the unit ideal. Conclusion: we have lowered the level.

It remains to make explicit the action of T, as alluded to above. Let X_{p}(J) = Hom(T , G_{m}) be the character group of the toric part T of the special fiber of J.

• X_{p}(J) supports a functorial action of T, since T preserves the canonical decomposition (7.1).

• X_{p}(J) supports an action of Gal(\overline{F}_{p}/F_{p}) which factors through a quadratic

extension of F_{p}. Think of the Galois action as an action of Frob_{p} ∈ D_{p} ⊂ Gal(\overline{Q}_{p}/Q_{p}). The action is at worst quadratic for the same reason that Tate curves have split multiplicative reduction after passing to a quadratic extension. On X_{p}(J) we have

(7.2)  T_{p} = Frob_{p},
(7.3)  T_{p}^{2} = ⟨p⟩.
Dually, we find T = Hom_{Z}(X_{p}(J), G_{m}) and
V ⊂ T (\overline{F}_{p})[ℓ] = Hom_{Z}(X_{p}(J), μ_{ℓ}).

Since Frob_{p} acts as p on μ_{ℓ}, we find that Frob_{p} acts as p∘T_{p} on T (\overline{F}_{p}). In particular, Frob_{p} acts on V ⊂ T (\overline{F}_{p}) as pa_{p} ∈ T/𝔪, i.e., as a scalar. Because Frob_{p} acts as a scalar and det(ρ) = χε, we have simulatenously

det(ρ(Frob_{p})) = \begin{cases}pε(p) & and \\ (pa_{p})^{2}. \end{cases}

But on V we have, by (7.3), that a_{p}^{2} = T_{p}^{2} = ⟨p⟩ = ε(p), so p^{2} ≡ p (mod ℓ). Since p ≠ ℓ, this can only happen if p ≡ 1 (mod ℓ).

Having made explicit the action of T on the torus, we have verified that both conditions hold, and in the Mazur case we can lower the level.

LECTURE 8

Level lowering with multiplicity one

Believe it or not, we assume that M = K · q with q ∤ K and that ε is really defined mod K, so that ρ arises from a from on Γ_{1}(K) ∩ Γ_{0}(pq). Make the

Key assumption: ρ is ramified at q.

The trick is to use q as a pivot. Edixhoven pointed out to me the possibility of doing this, in the context of Fermat. Associated to a (hypothetical) solution a^{ℓ} + b^{ℓ} + c^{ℓ} = 0 is a modular Galois representation V = E[ℓ] attached to an elliptic curve. Using Tate’s algorithm one discovers that Δ_{E} = \frac{(abc)^{2ℓ}}{2^{8}} which is a perfect ℓ-th power away from 2, and that the conductor is N_{E} = rad(abc). Take q = 2. Then E[ℓ] is definitely ramified at 2. Use 2 as a pivot to successively remove primes dividing the level N_{E}. So in real life^{1} this setup can occur, though one doesn’t always have this luxury. To complete Fermat, use q = 2 to remove each odd factor from N. It is certainly plausible (second case) that ℓ | N. Still, in this case k(ρ) = 2, so use [17] to lower the weight from ℓ + 1 to 2.

How do we get going? There is an unavoidable construction: Shimura curves. Denote by X(K, pq) the modular curve associate to Γ_{1}(K) ∩ Γ_{0}(pq) and let J = Jac(X(K, pq)) be its Jacobian. Likewise, denote by X^{pq}(K) the Shimura curve associated to the quaternion algebra of discriminant pq. How does this work?^{2} Let B be an indefinite quaternion algebra over Q of discriminant pq. (Up to isomorphism, B is unique.) Let O be an Eichler order of level K (i.e., reduced discriminant Kpq) in B. Let Γ_{∞} be the group of elements of O with (reduced) norm 1. After fixing an embedding B → M(2, R) (which exists because B is indefinite), we obtain in particular an embedding Γ_{∞} ↪ SL(2, R) and therefore an action of Γ_{∞} on the Poincaré upper half-plane H. Let X^{pq}(K) be the standard canonical model, over Q, of the compact Riemann surface Γ_{∞}\H, and let J′ = Jac(X^{pq}(K)) denote its Jacobian. The curve X^{pq}(K) is furnished with Hecke correspondences T_{n} for n ≥ 1. We write T_{n} for the endomorphism of J induced by the T_{n} via Pic functoriality.

By work of [18, 21, 43] there is a correspondence

[figure: traces of automorphic forms; traces of cusp forms]

1Brian Conrad remarks that counterexamples to Fermat are not real life.

2The following is essentially copied from your Invent. 100 paper.

Upon understanding this, and applying Faltings’s isogeny theorem [19], we find (noncanonically!) an isogeny J′ → J = Jac(X(K, pq)). The kernel is (of course) finite and the image is the pq-new part of J.

Amazingly, there seems to be no canonical map J′ → J. Amazingly, you do seem some canonical relationship between J′ and J: after looking at the corresponding character groups. But, there is a “pq switch.” The Cerednik-Drinfeld theory [8, 14] gives a description of X^{pq}(K) in characteristic p. Using this one finds that the character groups are related as follows: there is a canonical T-equivariant exact sequence

0 ⟶ X_{p}(J′) ⟶ X_{p}(J)X_{q}(J′′) ⟶ 0 (8.1)

where J′′ = Jac(X(K, q))^{×2}. Amazingly, this exact sequence relates a character groups “in characteristic p” to a character group “in characteristic q.” The Eichler order O has two residue fields of orders p and q.

Since ρ is ramified at q, 𝔪 ⊂ T is not q-old. We may as well suppose we are in a situation where we can’t lower the level using Mazur’s principle, so we assume that 𝔪 is not p-old either (and hope for a contradiction). Then, upon localizing (8.1) at 𝔪, the cokernel goes away and we find that

X_{p}(J)/𝔪X_{p}(J) ≈ X_{q}(J′)/𝔪X_{q}(J′) (8.2)

and vice versa, i.e.,

X_{q}(J)/𝔪X_{q}(J) ≈ X_{p}(J′)/𝔪X_{p}(J′). (8.3)

Each of these is an isomorphism of T/𝔪-vector spaces. Recall [3] that J[𝔪] ≈ ⨁_{i=1}^{λ} V, with λ > 0 and J′[𝔪] ≈ ⨁_{i=1}^{ν} V. It follows from [21] that ν > 0, but we will not use this. Our hypothesis that V is unramified automatically propagates to all of J[𝔪] = ⊕V. Since V is irreducible and we are assuming that 𝔪 is not p-old, the same argument as in the previous lecture shows that J[𝔪] ⊂ T [𝔪] where T is the toric part of J_{F_{p}}. This means^{3} that dim(X_{p}(J)/𝔪X_{p}(J)) ≥ 2λ. Using the same argument with J replaced by J′ gives that dim(X_{p}(J′)/𝔪X_{p}(J′)) ≥ 2μ.

Since V is ramified at q (by hypothesis) and there is an unramified line (?), dim(V^{I_{q}}) = 1. Then dim J[𝔪]^{I_{q}} = λ, which since Hom(X_{q}(J)/𝔪X_{q}(J), μ_{ℓ}) ⊂ J[𝔪]^{I_{q}} bounds the dimension of X_{q}(J)/𝔪X_{q}(J). After applying a similar argument with X_{q}(J′) and recalling the previous paragraph, we obtain the following quintuple of inequalities:

dim X_{q}(J)/𝔪X_{q}(J) ≤ λ
dim X_{q}(J′)/𝔪X_{q}(J′) ≤ μ
dim X_{p}(J)/𝔪X_{p}(J) ≥ 2λ
dim X_{p}(J′)/𝔪X_{p}(J′) ≥ 2μ

Combining these with (8.2, 8.3) we find that

2λ ≤ dim X_{p}(J)/𝔪X_{p}(J)
= dim X_{q}(J′)/𝔪X_{q}(J′) ≤ μ

3Naive question: Ken, would failure of multiplicity one (in an interesting case) evidence itself in the T-module X_{p}(J) behaving in a way completely different than H_{1}(X_{0}(N), Z)?

and simulatenously that 2μ ≤ λ. Combining gives that 4λ ≤ λ, so λ = 0, which is contrary to fact: the multiplicity of ρ in J[𝔪] is strictly positive.

This completes the proof, and the lecture series.

Question: What is the next hard case? It is illustrated by the following diagram:

[figure: Mpq; pivot; Mq; easy; Mp; level addition; M]

The pivot step is potentially the hardest. This is because the symmetry is broken. Before we knew that q can’t be subtracted from the level. But here, q can be subtracted. The argument has been made to work [34] when ℓ > 2.

LECTURE 9

Katz modular forms (Appendix)

In this section we briefly introduce modular forms modulo ℓ from a geometric point of view^{1} Our purpose is to give a brief overview, rather than a complete (or even well motivated) account.

Modular forms as rules. Fix a base ring R. Consider pairs (E, ω)/S where S is a ring over R, E is an elliptic curve over S enhanced with level structure, and ω is a differential which does not vanish modulo any prime ideal of S. A modular form f of weight k is a certain type of rule which takes any such pair (E, ω)/S to an element of S. The two main properties of f are:

1. Weight: For all λ ∈ R^{∗}, f(E, λω) = λ^{−k}f(E, ω).

2. Holomorphicity: f commutes with base change in the evident sense.

3. Tate curves: Condition to insure that q-expansions do not have negative powers of q.

This definition is difficult to think about because we have only

(rules) + (functoriality)

A priori, we do not know what is going on. Fortunately, we know what elliptic curves are, not in the sense of Shimura-Taniyama, but in the moduli sense. Supposing N ≥ 5, there exists a model for X_{1}(N) over Z[1/N] and a scheme E_{1}(N) over X_{1}(N) such that for all rings R in which N is invertible, and generalized elliptic curves E enhanced with level structure, there is a unique map Spec(R) → X_{1}(N) making the following diagram Cartesian, i.e., a fiber product:

\xymatrix{E \ar[r] \ar[d] & E_{1}(N) \ar[d] \\ Spec(R) \ar[r] & X_{1}(N)}

Glueing together differentials on each elliptic curve E corresponding to a point of X_{1}(N) gives a line bundle \underline{ω} on X_{1}(N). Weight k modular forms on Γ_{1}(N)/Z[1/N] are the same thing as elements of H^{0}(X_{1}(N), \underline{ω}^{⊗k}). This is a finite module over Z[1/N] which gives all rules of weight k and level N.

1This section records a lecture of Mathew Emerton. This lecture should be an appendix, and might be replaced/supplemented by notes that Brian Conrad is writing up on his lectures.

Modular forms mod ℓ. Suppose ℓ is a prime with ℓ ∤ N. The space of weight k modular forms on Γ_{1}(N)/F_{ℓ} is the same as

H^{0}(X_{1}(N)/F_{ℓ}, \underline{ω}^{⊗k}).

To relate this space to modular forms over Z[1/N] first consider the diagram:

[figure: E/F_{ℓ}; E/Z[1/N]; \underline{ω}/F_{ℓ}; \underline{ω}/Z[1/N]; X_{1}(N)/F_{ℓ}; X_{1}(N)/Z[1/N]; Spec(F_{ℓ}); Spec(Z[1/N])]

Do all of the modular forms over F_{ℓ} come via reduction from modular forms over Z[1/N]? Yes, provided that the appropriate cohomology commutes with base change. This is a subtle question because, in general, cohomology need not commute with base change. Via some trickery, and consideration of the exact sequence

0 → \underline{ω}^{⊗k} ⟶^{ℓ} \underline{ω}^{⊗k} ⟶ \underline{ω}^{⊗k}/F_{ℓ} → 0

one sees that a condition which insures that all forms come from characteristic 0 is that

H^{1}(X_{1}(N)/F_{ℓ}, \underline{ω}^{⊗k}) = 0.

The Riemann-Roch theorem often gives this vanishing, the essential ingredient being knowledge of deg(\underline{ω}). For example, when k ≥ 2 and N ≥ 5, the H^{1} does vanish! There are problems when N ≤ 5 because an appropriate representing object E_{1}(N) → X_{1}(N) does not exist.

Flat base change commutes with cohomology. For example Z[1/N] ↪ C is flat because C is torsion free, whereas Z[1/N] → F_{ℓ} is not flat. Observe that H^{0}(X_{1}(N), ω^{⊗k}/Z[1/N]) is just a choice of Z[1/N] lattice in our favorite complex vector space of cusp forms. The following diagram is very helpful (forgetting it is a common source of confusion.)

\xymatrix{C ⊕ ⨁_{k≥2} M_{k}(C) \ar@{^{(}->}[rr] &  & C[[q]] \\ Z[\frac{1}{N}] ⊕ ⨁_{k≥2} M_{k}(Z[\frac{1}{N}]) \ar@{^{(}->}[rr] \ar@{^{(}->}[u] \ar[d] &  & (Z[\frac{1}{N}])[[q]] \ar@{^{(}->}[u] \ar[d] \\ F_{ℓ} ⊕ ⨁_{k≥2} M_{k}(F_{ℓ}) & not injective \ar[r] & F_{ℓ}[[q]]}

There exists modular forms over F_{ℓ} of different weights which have the same q-expansions. Consider the Hasse invariant A ∈ H^{0}(X_{1}(N)/F_{ℓ}, ω^{⊗ℓ−1}). Let A/F_{ℓ} be an elliptic curve and let Φ denote the Frobenius endomorphism. The map H^{1}(E, O) → H^{1}(E, O) induced by Frobenius is either 0 or not; if not [[fill in details here.]] The q-expansion of A is 1 ∈ F_{ℓ}[[q]]. This is the same as the q-expansion of 1.

In characteristic 0, weight 1 forms are very special so Serre didn’t allow them in his conjecture. In characteristic p, weight 1 forms behave in pretty much the same way as other weights.

The references are: Serre, Bourbaki, 1969 Serre, Swinnerton-Dyer, LNM 350 “Galois representations” Katz “very geometric” Gross “modular forms mod ℓ” Edixhoven Jochnowitz, Duke, Trans. of AMS

LECTURE 10 Exercises (Appendix)

These are the exercises^{1} which were used in the problem sessions. Some solution are provided at the end. [Todo: change the E, F ’s to A, B’s to be consistent with Ken’s notation in the lectures.]

Exercise 10.1. Suppose ρ : Gal(\overline{Q}/Q) → F_{ℓ}^{∗} is a one dimensional continuous odd Galois representation.

1. Give an example to show that ρ need not be a power of the mod ℓ cyclotomic character.

2. Assume that ρ is unramified outside ℓ. Deduce that ρ is a power of the cyclotomic character.

Exercise 10.2. The principal congruence subgroup Γ(N) of level N is the subgroup of SL_{2}(Z) consisting of matrices congruent to the identity modulo N. Further, Γ_{1}(N) is the subgroup of SL_{2}(Z) of matrices which are upper triangular mod N and have diagonal entries ≡ 1 mod N. A congruence subgroup is a subgroup of SL_{2}(Z) that contains Γ(N) for some N. Let Γ ⊂ SL_{2}(Z) be a congruence subgroup. Show that there exits g ∈ GL_{2}(Q) such that g^{−1}Γg ⊂ GL_{2}(Q) contains Γ_{1}(N) for some N.

Exercise 10.3. Let E/Q be an elliptic curve. Show that End_{Q}(E) = Z, that is, integer multiplications are the only Q-rational endomorphisms of E. Assume further that E is “isolated in its isogeny class,” i.e., if F/Q is an elliptic curve which is isogeneous to E over Q, then E and F are isomorphic over Q. Show that, for every prime number ℓ, the representation

ρ_{ℓ} : Gal(\overline{Q}/Q) → Aut(E[ℓ]) ≈ GL(2, F_{ℓ})

is irreducible.

Exercise 10.4. Let E/Q be an elliptic curve, and assume that, for all ℓ, the representation ρ_{E,ℓ} : Gal(\overline{Q}/Q) → Aut(E[ℓ]) is irreducible. Show that E is isolated in its isogeny class.

Exercise 10.5. Suppose ρ : Gal(\overline{Q}/Q) → GL_{2}(F_{ℓ}) arises from the ℓ-torsion of an elliptic curve. Verify, using standard properties of the Weil pairing, that det(ρ) is the mod ℓ cyclotomic character.

1The authors would like to thank D. Savitt, K. Kedlaya, and B. Conrad for contributing problems.

Exercise 10.6. Let f ∈ S_{k}(Γ_{1}(N)) be a modular form which is an eigenform for all of the Hecke operators T_{p} and for the diamond bracket operators ⟨d⟩. Let

ε : (Z/NZ)^{∗} → C^{∗}

be the character of f, so ⟨d⟩f = ε(d)f.

1. Show that f satisfies the following functional equation:

for any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),

f(z) = ε(d)(cz + d)^{−k}f(\frac{az + b}{cz + d}).

2. Conclude that ε(−1) = (−1)^{k}.

3. Choose a mod ℓ Galois representation ρ associated to f. Its determinant is det(ρ) = ε · χ^{k−1} where χ is an appropriate cyclotomic character. If c is a complex conjugation, deduce that that ρ is odd:

det(ρ(c)) = −1.

4. Assume ℓ > 2. Prove that ρ(c) is conjugate over \overline{F}_{ℓ} to the matrix ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ). What happens if ℓ = 2.

Exercise 10.7. Construct a non-continuous homomorphism

ρ : Gal(\overline{Q}/Q) → \{±1\}

where \{±1\} has the discrete topology. You must produce a map ρ : Gal(\overline{Q}/Q) → {±1} such that

1. ρ is a homomorphism, and

2. ρ does not factor through Gal(K/Q) for any finite Galois extension K/Q.

Exercise 10.8. One difficulty is that the representation ρ arising from a modular form sometimes takes values in a slightly smaller field than O/λ. For example, let f be one of the two conjugate newforms of level 23, weight 2, and trivial character. Then

f = q + αq^{2} + (−2α − 1)q^{3} + (−α − 1)q^{4} + 2αq^{5} + ⋯

with α^{2} + α − 1 = 0. The coefficients of f lie in O = Z[α] = Z[\frac{1+\sqrt{5}}{2}]. Take λ to be the unique prime of O lying over 2, then O/λ ≅ F_{4} and so \overline{ρ}_{f,λ} is a homomorphism to GL(2, F_{4}). Show that if p ≠ 2 then a_{p} ∈ Z[\sqrt{5}], so that \overline{ρ}_{f,λ} has a model over GL(2, F_{2}).

Exercise 10.9. Let E/Q be an elliptic curve and ℓ ≠ 2 a prime.

1. Prove that the field Q(E[ℓ]) generated by the coordinates of the points in E[ℓ] is not equal to Q.

2. Given an example of an elliptic curve E so that Q(E[2]) = Q.

Exercise 10.10. Let E be the elliptic curve defined by

y^{2} = x^{3} + ax + b with a, b ∈ Q.

1. Describe the Galois representation

ρ = ρ_{E,2} : Gal(\overline{Q}/Q) ⟶ GL(2, F_{2}).

2. Give necessary and sufficient conditions for ρ to be reducible.

3. Given an example in which ρ is ramified at 23.

Exercise 10.11. Let ε and ρ be continuous homomorphisms ε, ρ : Gal(\overline{Q}/Q) → F_{ℓ}^{∗}. Suppose further that for all primes p, for which both ρ and ε are unramified, we have

ρ(Frob_{p}) = ε(Frob_{p})p^{i} ∈ F_{ℓ}.

Deduce that ρ = ε · χ^{i} where χ is the mod ℓ cyclotomic character.

Exercise 10.12. Let E/Q be an elliptic curve of conductor N. Let p be a prime number not dividing N. Denote by Ẽ the elliptic curve E considered over the finite field F_{p}. There is a Frobenius endomorphism Φ = Φ_{p} : E → E which sends an affine point (x, y) to the point (x^{p}, y^{p}) and fixes ∞. Let X^{2} + aX + b be the characteristic polynomial of Φ acting on the Tate module of E, for some (any) prime ℓ ≠ p. Define tr(Φ) = −a and deg(Φ) = b.

1. Show that deg(Φ).

2. Show that tr(Φ) = p + 1 − #E(F_{p}), i.e., tr(Φ) = a_{p}.

3. Let ℓ ∤ pN be a prime. Then Ẽ[ℓ] is a vector space of dimension two over F_{ℓ}, and Φ induces a map Ẽ[ℓ] → Ẽ[ℓ]. Show that this is exactly the same map as that induced by some choice of Frob_{p} ∈ Gal(\overline{Q}/Q).

4. Conclude that

tr(ρ_{E,ℓ}(Frob_{p})) = p + 1 − #E(F_{p}) (mod ℓ).

Exercise 10.13. Suppose E/Q is an elliptic curve of conductor N, and let ℓ be a prime. Show that any prime p not dividing ℓN is unramified in Q(E[ℓ])/Q. You may use the following fact which is proved using formal groups: The map E[ℓ] → Ẽ(\overline{F}_{p}) is injective, where Ẽ is the reduction of E modulo p. (Using formal groups one can show that all torsion in the kernel is of p-power order.

Exercise 10.14. Show that the fundamental character of level 1 is the cyclotomic character χ|_{I}.

Exercise 10.15. Let ρ : I_{t} → GL(2, \overline{F}_{ℓ}) be a representation, where I_{t} is the tame inertia group at ℓ. Show that ρ is diagonalizable.

Exercise 10.16. Guess the Eichler-Shimura embedding

S_{2}(Γ_{1}(N)) ↪ H^{1}(Γ_{1}(N), C)

in weight 2.

Exercise 10.17. For each of the following semistable elliptic curves A, and each ℓ at which ρ_{A,ℓ} is irreducible, compute the Serre minimal weight k(ρ_{A,ℓ}) and level N(ρ_{A,ℓ}).

N  |Δ|  reducible ℓ A
30  2^{4} · 3^{5} · 5  2, 3  y^{2} + xy + y = x^{3} + x + 2
210  2^{12} · 3^{3} · 5 · 7  2, 3  y^{2} + xy = x^{3} − 41x − 39
330 2^{4} · 3^{2} · 5^{4} · 11^{2}  2  y^{2} + xy = x^{3} + x^{2} − 102x + 324
455  5^{3} · 7^{4} · 13  2  y^{2} + xy = x^{3} − x^{2} − 50x + 111
2926 2^{8} · 7^{3} · 11^{4} · 19^{2}  2  y^{2} + xy + y = x^{3} − x^{2} + 1934x − 1935

Exercise 10.18. Let M be a positive integer and p a prime. Show that there is injective linear map

S_{2}(Γ_{1}(M)) ↪ S_{2}(Γ_{1}(pM))

sending f(q) to f(q^{p}).

Exercise 10.19. Let M be such that S_{2}(Γ_{1}(M)) has positive dimension and let p be a prime.^{2}

1. Let f ∈ S_{2}(Γ_{1}(M)) be an eigenvector for T_{p} with eigenvalue λ. Show that T_{p} acting on S_{2}(Γ_{1}(Mp)) preserves the two dimensional subspace generated by f and T_{p}f. Show furthermore that if λ^{2} ≠ 4p^{k−1} then T_{p} is diagonalizable on this 2-dimensional space. What are the eigenvalues of T_{p} on this space? Remark: I believe there is no example known where λ^{2} = 4p^{k−1} and it’s a theorem that it can never happen if k = 2.

2. Show that for any r > 2, the Hecke operator T_{p} on S_{2}(Γ_{1}(Mp^{r})) is not diagonalizable. Conclude that the Hecke algebra T associated to S_{2}(Γ_{1}(Mp^{r})) can not be an order in a product of rings of integers of number fields.

Exercise 10.20. Let N be a positive integer. Show that the Hecke algebra T = Z[… T_{n} …] ⊂ End_{Q}(J_{1}(N)) is of finite rank as a Z-module.

Exercise 10.21. Let Spec(T) be the affine scheme associated to the Hecke algebra T. Give a careful interpretation of the geometry of Spec(T) in terms of eigenforms and Galois representations. Draw pictures to illustrate your ideas.

Exercise 10.22. Suppose N = pM with (p, M) = 1. There is an injection

S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)) ↪ S_{2}(Γ_{1}(N) ∩ Γ_{0}(p))

given by (f, g) ↦ f(q)+g(q^{p}). The Hecke algebra T = T_{N} acts through a quotient \overline{T} on the image of S_{2}(Γ_{1}(M)) ⊕ S_{2}(Γ_{1}(M)). Suppose 𝔪 ⊂ T is a maximal ideal which arises by pullback from a maximal ideal in \overline{T}. Show that ρ_{𝔪} arises from a modular form of level M.

Exercise 10.23. An exercise on Tate curves.

Solutions.

Solution 10.1. 1. Let p be a prime different than ℓ and

ρ : Gal(\overline{Q}/Q) → Gal(Q(\sqrt{p})/Q) = \{±1\} ↪ F_{ℓ}^{∗}.

2. Let K = Q^{ker(ρ)}. Then K/Q is abelian and ramified only at ℓ, so K ⊂ Q(ζ_{ℓ^{∞}}). But [K : Q] | ℓ − 1 so K ⊂ Q(ζ_{ℓ}).

Solution 10.2. Conjugate using g = ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) .

Solution 10.3. Suppose φ ∈ End_{Q}(E) is a nonzero endomorphism. The induced map dφ on the differentials H^{0}(E, Ω) ≈ Q is multiplication by an integer n. Then d(φ − n) = 0 so φ = n.

Suppose that ρ_{ℓ} is reducible, so that there is a one dimensional Galois stable subspace V ⊂ E[ℓ]. The quotient F = E/V is then an elliptic curve over Q and there is an isogeny π : E → F of degree ℓ. By assumption that E is isolated in its isogeny class, we have in fact that F = E and so there is an endomorphism of E of degree ℓ. But E does not have CM so the endomorphism all have degree a perfect square.

2This problem is based on a Cambridge handout of Kevin Buzzard’s.

Solution 10.4. Suppose all ρ_{E,ℓ} are irreducible, yet there exists an isogeny φ : E → F with F ≅ E̸. Choose φ to have minimal possible degree and let d = deg(φ) > 1. Let ℓ be the smallest prime divisor of d and choose a point x ∈ ker(φ) of exact order ℓ. If the order ℓ cyclic subgroup generated by x is Galois stable, then ρ_{E,ℓ} is reducible, contrary to assumption. Thus ker(φ) contains the full ℓ-torsion subgroup E[ℓ] of E. In particular, φ factors as

\xymatrix{E \ar[rr]^{φ} \ar[dr]^{ℓ} &  & F \\  & E/E[ℓ] \ar[ur]}

Since E/E[ℓ] ≅ E, there is an isogeny from E to F of degree equal to d/ℓ^{2}, which is contrary to our assumption that d is minimal.

Solution 10.14. The level one character Ψ is obtained by the composition

Gal(Q_{ℓ}^{nr}(ℓ^{ℓ−1})/Q_{ℓ}^{nr}) → μ_{ℓ−1}(\overline{Q}_{ℓ}) → μ_{ℓ−1}(\overline{F}_{ℓ}^{∗}) = F_{ℓ}^{∗}.

Let π be such that π^{ℓ−1} = ℓ. Then Ψ(σ) = \frac{σ(π)}{π} (mod π). Let ζ ∈ \overline{Q}_{ℓ} be a primitive ℓth root of unity. Now

∏_{a=1}^{ℓ−1}(ζ^{a} − 1) = ℓ,

so

(ζ − 1)^{ℓ−1} ∏_{a=1}^{ℓ−1} \frac{ζ^{a} − 1}{ζ − 1} = ℓ

and (this is where Wilson’s theorem is used),

∏_{a=1}^{ℓ−1} \frac{ζ_{ℓ}^{a} − 1}{ζ − 1} ≡ 1 (mod ζ − 1).

Since the polynomial x^{ℓ−1} − 1 has roots over F_{ℓ}, by Hensel’s lemma there is a unit u ∈ Q_{ℓ}(ℓ^{ℓ−1}) such that

u^{ℓ−1} = ∏_{a=1}^{ℓ−1} \frac{ζ^{a} − 1}{ζ − 1}.

We can take π = (ζ − 1)u. Then

\frac{σ(π)}{π} = \frac{(ζ^{χ(σ)} − 1)σ(u)}{(ζ − 1)u}
= \frac{(ζ − 1)(ζ^{χ(σ)−1} + ⋯ + 1)σ(u)}{(ζ − 1)u}
= (ζ^{χ(σ)−1} + ⋯ + 1)σ(u)/u
≡ χ(σ) (mod ζ − 1)

Solution 10.5. The Weil pairing ( , ) : E[ℓ] × E[ℓ] → μ_{ℓ} can be viewed as a map

Λ^{2}E[ℓ] ⟶^{≅} μ_{ℓ}

sending P ∧ Q to (P, Q). For any σ ∈ Gal(\overline{Q}/Q), (P^{σ}, Q^{σ}) = (P, Q)^{σ}. With the action (P ∧ Q)^{σ} = P^{σ} ∧ Q^{σ}, the map Λ^{2}E[ℓ] → μ_{ℓ} is a map of Galois modules. To compute det(ρ(σ)) observe that if e_{1}, e_{2} is a basis for E[ℓ], and ρ(σ) = ( \begin{matrix}a & b \\ c & d \end{matrix} ), then

σ(e_{1} ∧ e_{2}) = (ae_{1} + ce_{2}) ∧ (be_{1} + de_{2})
= (ad − bc)e_{1} ∧ e_{2} = det(ρ(σ))e_{1} ∧ e_{2}

Thus Λ^{2}E[ℓ] gives the one dimensional representation det(ρ). Since Λ^{2}E[ℓ] is isomorphic to μ_{ℓ} it follows that det(ρ) = χ.

Solution 10.6. The definition of ⟨d⟩ is as follows: choose any matrix σ_{d} ∈ Γ_{0}(N) such that σ_{d} ≡ ( \begin{matrix}d & 0 \\ 0 & d^{−1} \end{matrix} ) (mod N), then ⟨d⟩f = f|_{σ_{d}}. Observe that Γ_{1}(N) is a normal subgroup of Γ_{0}(N) and the matrices σ_{d}, for (d, N) = 1, d < N, are a system of coset representatives. Thus any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) can be written in the form σ_{d} · g for some g ∈ Γ_{1}(N). We have

f = f|_{σ_{d}g} = (f|_{σ_{d}})|_{g} = (ε(d)f)|_{g} = ε(d)f|_{g} = ε(d)(cz + d)^{−k}f (\frac{az + b}{cz + d}) .

Solution 10.6.4. Let y^{2} = x(x^{2} − a) with a ∈ Q not a square. Then

A[2] = \{∞, (0, 0), (\sqrt{a}, 0), (−\sqrt{a}, 0)\}.

The action of c on the basis (0, 0), (\sqrt{a}, 0) is represented by the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), since c(−\sqrt{a}, 0) = (\sqrt{a}, 0) = (0, 0) + (−\sqrt{a}, 0).

Solution 10.7.

1. Choose a sequence p_{1}, p_{2}, … of distinct prime numbers. Define ρ_{1} : G → ∏ F_{2} by

ρ_{1}(σ)_{i} = \begin{cases}0 & if σ acts trivially on Q(\sqrt{p_{i}}), \\ 1 & otherwise \end{cases}

Thus ρ_{1} is just

G → Gal(Q(\sqrt{p_{1}}, \sqrt{p_{2}}, …)/Q) ≈ ∏ F_{2}.

2. Let ⊕F_{2} ⊂ ∏ F_{2} be the subgroup of elements having only finitely many nonzero coordinates. Then ∏ F_{2}/⊕F_{2} is a vector space over F_{2} of dimension > 0. By Zorn’s lemma, there is a basis B of ∏ F_{2}/ ⊕ F_{2}. Let b ∈ B and and let W be the space spanned by B − {b}. Then V = (∏ F_{2}/ ⊕ F_{2})/W is an F_{2} vector space of dimensional 1.

3. Let ρ be the composite map

\xymatrix{Gal(\overline{Q}/Q) \ar[d] \ar[drr]^{ρ} \\ ∏ F_{2} \ar[r] & V \ar[r]_{≅} & \{±1\}}

4. Let H = ker(ρ) ⊂ Gal(\overline{Q}/Q). If σ(\sqrt{p_{i}}) = −\sqrt{p_{i}} and σ(\sqrt{p_{j}}) = \sqrt{p_{j}} for i ≠ j, then σ ∈ H. Thus H does not fix Q(\sqrt{p_{i}}), so the fixed field of H is Q. Thus the largest quotient through which ρ might factor is Gal(Q/Q) = {1}; but ρ visibily does not factor through the trivial group.

Solution 10.8. We have f = f_{1} + αf_{2} with

f_{1} = q − q^{3} − q^{4} + ⋯
f_{2} = q^{2} − 2q^{3} − q^{4} + 2q^{5} + ⋯

Because S_{2}(Γ_{0}(23)) has dimension 2, it is spanned by f_{1} and f_{2}. Let η(q) = q^{\frac{1}{24}} ∏_{n≥1}(1 − q^{n}). Then g = (η(q)η(q^{23}))^{2} ∈ S_{2}(Γ_{0}(23)). Expanding we find that g = q^{2} − 2q^{3} + ⋯, so g = f_{2}. Next observe that g is a power series in q^{2}, modulo 2, since

g = q^{2} ∏(1 − q^{n})^{2}(1 − q^{23n})^{2}
≡ q^{2} ∏(1 − q^{2n})(1 − q^{46n}) (mod 2)
≡ q^{2} ∏(1 + q^{2n} + q^{46n} + q^{48n}) (mod 2)

Thus the coefficient in f_{2} of q^{p}, with p ≠ 2 prime, is even and the proposition follows.

Solution 10.9.

1. Let ζ ∈ μ_{ℓ} be a primitive ℓth root of unity. Since Λ^{2}E[ℓ] ≅ μ_{ℓ}, there exists P, Q ∈ E[ℓ] such that P ∧Q = ζ. Since ℓ > 2 there exists σ such that ζ^{σ} ≠ ζ, hence P^{σ} ∧ Q^{σ} ≠ P ∧ Q. This is impossible if all ℓ-torsion is rational, since then P^{σ} = P and Q^{σ} = Q.

2. y^{2} = (x − a)(x − b)(x − c) where a, b, c are distinct rational numbers.

Solution 10.10.

1. Let K be the splitting field of x^{3} + ax + b. Then ρ embeds Gal(K/Q) in GL(2, F_{2}):

\xymatrix{Gal(\overline{Q}/Q) \ar[rr]^{ρ} \ar[dr] &  & GL(2, F_{2}) \\  & Gal(K/Q) \ar@{^{(}->}[ur]}

2. ρ is reducible exactly when the polynomial x^{3} + ax + b has a rational root.

3. y^{2} = x(x^{2} − 23), y^{2} = x^{3} + x − 1. [[Insert Ravi’s geometric comments here.]]

Solution 10.11. Consider the character τ = εχ/ρ. By assumption τ(Frob_{p}) = 1 for all unramified p. Furthermore, τ factors through a finite abelian extension of K/Q. For any σ ∈ Gal(K/Q), the Cebotarev density theorem insures that there are infinitely many primes p such that Frob_{p} = σ. In particular, τ(σ) = τ(Frob_{p}) = 1, so τ = 1 and hence ρ = εχ.

Solution 10.12.

1. deg(Φ) = p, [45, 2.11].

2. By [45, 5.5], Φ−1 is separable, so #E(F_{p}) = deg(Φ−1). Since Φ has degree p, there exists an isogeny \overline{Φ} (the dual isogeny [45, III.6]), such that Φ\overline{Φ} = p.

Letting bars denote the dual isogeny, we have

#E(F_{p}) = deg(Φ − 1) = (Φ − 1)\overline{(Φ − 1)}
= Φ\overline{Φ} − Φ − \overline{Φ} + 1
= p − tr(Φ) + 1

3. Both maps are pth powering on coordinates.

Solution 10.13. Because ℓ ≠ p and E has good reduction at p, the natural map E[ℓ] → Ẽ[ℓ] is an isomorphism. We have the following commutative diagram

\xymatrix{Gal(Q_{p}(E[ℓ])/Q_{p}) \ar@{^{(}->}[r] \ar[d] & Aut(E[ℓ]) \ar[d]^{≅} \\ Gal((O/λ)_{F_{p}}) \ar[r] & Aut(Ẽ[ℓ])}

It follows that the first vertical map must be injective, which is the same as Q_{p}(E[ℓ])/Q_{p} being unramified.

Solution 10.15. I_{t} is an abelian profinite group which, by Kummer theory, has order prime to ℓ. If σ ∈ I_{t}, then the matrix ρ(σ) has Jordan Canonical Form ( \begin{matrix}a & b \\ 0 & c \end{matrix}). If b ≠ 0 then a = c, and the matrix is ( \begin{matrix}a & b \\ 0 & a \end{matrix}), which has order divisible by ℓ, which contradicts the fact that the elements of I_{t} have order prime to ℓ. Thus each ρ(σ) is diagonalizable. Now use that I_{t} is abelian and a commuting family of diagonalizable operators is simultaneously diagonalizable.

Solution 10.16. Since Γ_{1}(N) acts trivially on C, we have H^{1}(Γ_{1}(N), C) = Hom(Γ_{1}(N), C). Pick a base point z_{0} in the upper half plane. The element of H^{1} associated to f is then the map

γ ↦ ∫_{z_{0}}^{γ(z_{0})} f(z)dz.

Solution 10.17. We write N = N(ρ) and k = k(ρ) to save space. The essential tool is Theorem 5.1.

1. ℓ = 5: N = 6, k = 6, ℓ > 5, N = 30, k = 2.

2. ℓ = 5: N = 2 · 3 · 7, k = 6; ℓ = 7: N = 2 · 3 · 5, k = 8; ℓ > 7: N = 2 · 3 · 5 · 7,

k = 2.

3. ℓ = 3: N = 2·5·11, k = 4; ℓ = 5: N = 2·3·11, k = 6; ℓ = 7: N = 2·3·5·11,

k = 2; ℓ = 11: N = 2 · 3 · 5, k = 12; ℓ > 11: N = 2 · 3 · 5 · 11, k = 2.

4. ℓ = 3: N = 7 · 13, k = 2; ℓ = 5: N = 7 · 13, k = 6; ℓ = 7: N = 5 · 13, k = 8;

ℓ = 13: N = 5 · 7, k = 14; ℓ = 11, ℓ > 13: N = 5 · 7 · 13, k = 2.

5. ℓ = 3: N = 2·11·19, k = 2; ℓ = 7: N = 2·11·19, k = 8; ℓ = 11: N = 2·7·19,

k = 12; ℓ = 19: N = 2 · 7 · 11, k = 20; ℓ other: N = 2 · 7 · 11 · 19, k = 2.

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